#72287671

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂²×#41	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2220 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)(3,11)(4,15)(5,10)(6,9)(7,13)(8,17)(12,19)(14,24)(16,28)(18,22)(20,27)(21,29)(23,25)(26,30)(31,32)(34,40)(35,45)(36,47)(43,56)(44,57)(46,48)(50,51)(62,64);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,23)(16,26)(17,25)(18,27)(21,30)(22,24)(28,31)(29,32)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,54)(46,58)(47,55)(48,59)(49,57)(50,60)(51,61)(52,62)(53,56)(63,64);(1,4,5,16,6,17,18,29)(2,8,10,21,9,15,22,28)(3,12,14,26,13,25,27,32)(7,19,20,30,11,23,24,31)(33,35,37,46,39,45,53,48)(34,41,43,58,40,54,56,59)(36,38,50,52,47,49,51,63)(42,44,60,62,55,57,61,64);(3,13)(7,11)(12,25)(14,27)(19,23)(20,24)(26,32)(30,31)(34,40)(35,45)(36,47)(43,56)(44,57)(46,48)(50,51)(62,64);(2,9)(3,13)(8,15)(10,22)(12,25)(14,27)(21,28)(26,32)(33,36,49,48,37,51,63,45,39,47,38,46,53,50,52,35)(34,42,57,59,43,61,64,54,40,55,44,58,56,60,62,41)

Polycyclic presentation:

2^10|c^2=f,e^2=gj,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=ch,d^b=dh,d^c=dj,e^a=eh,e^b=ehj,e^c=eg,e^d=ej,f^e=fj,g^c=gi,i^c=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287792, #72287829, #72318707
maximal normal subgroup of	#72287792, #72287829, #72318707

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, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);