#72287698

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	#3444	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287792, #72318999, #72319199
maximal normal subgroup of	#72287792, #72318999, #72319199

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.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.10);;
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*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
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);