#72287309

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623517	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,9)(4,10)(5,20)(6,12)(7,11)(8,13)(14,23)(15,26)(16,25)(17,27)(18,28)(19,30)(21,24)(22,29)(31,32)(33,34,39,45)(35,40,46,41)(36,42,53,60)(37,43,52,55)(38,44,54,63)(47,56,49,58)(48,57,51,61)(50,59,64,62);(1,3)(2,9)(4,18)(5,21)(6,16)(7,15)(8,17)(10,28)(11,26)(12,25)(13,27)(14,19)(20,24)(22,31)(23,30)(29,32)(35,46)(40,41)(47,49)(48,51)(50,64)(56,58)(57,61)(59,62);(1,4,7,6,8,19,5,22)(2,10,11,12,13,30,20,29)(3,14,15,31,17,18,21,16)(9,23,26,32,27,28,24,25)(33,35,36,48,38,50,52,49,39,46,53,51,54,64,37,47)(34,40,42,57,44,59,55,58,45,41,60,61,63,62,43,56);(1,5,8,7)(2,11,13,20)(3,15,17,21)(4,6,19,22)(9,24,27,26)(10,29,30,12)(14,16,18,31)(23,32,28,25)(33,35,37,49,54,64,53,48,39,46,52,47,38,50,36,51)(34,41,43,56,63,59,60,61,45,40,55,58,44,62,42,57);(1,6,7,19,8,22,5,4)(2,12,11,30,13,29,20,10)(3,16,15,14,17,31,21,18)(9,25,26,23,27,32,24,28)(35,47,46,49)(36,52)(37,53)(38,54)(40,56,41,58)(42,55)(43,60)(44,63)(48,50,51,64)(57,59,61,62)

Polycyclic presentation:

2^10|a^2=j,c^2=f,d^2=g,e^2=fghij,f^2=h,g^2=ij,h^2=j,i^2=j,c^b=chi,d^a=dhi,d^b=dhi,d^c=dh,e^c=egij,e^d=egij,f^d=fj,f^e=fi,g^c=gj,g^e=gij,h^e=hj,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287390, #72287657, #72318672
maximal normal subgroup of	#72287390, #72287657, #72318672

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);