#72256102

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13956
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#7713863	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(5,8)(9,10)(11,19)(12,23,26,35)(13,18)(14,17)(15,22)(16,27,33,29)(20,37,24,40)(21,34)(25,32)(28,31)(30,39,36,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15,17,19)(12,24,26,20)(13,28)(14,22)(16,30)(18,34,32,31)(21,25)(23,37)(27,38,29,39)(33,36)(35,40);(1,3,4,7)(2,5)(6,8)(9,12,25,27,14,26,13,29)(10,16,32,23,17,33,18,35)(11,20,28,38,22,24,21,39)(15,30,34,40,19,36,31,37);(3,7)(5,8)(10,17)(12,26)(13,25)(15,19)(16,33)(20,24)(21,28)(30,36)

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=efi,e^2=ij,b^a=bg,c^a=ce,c^b=cghi,d^a=dij,d^c=dhj,e^b=ei,e^c=eij,e^d=eij,f^c=fij,g^b=gij,g^c=gj,h^a=hij,h^c=hij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256124, #72256215, #72256392
maximal normal subgroup of	#72256124, #72256215, #72256392

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