#72306758

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8522
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₄×M₁₆	C₂⁴
socle	C₂	C₂×#8522
prime radical	#10575509	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=e,e^2=hj,g^2=i,h^2=j,c^a=cf,c^b=cfj,d^b=dj,d^c=dg,e^a=ei,e^b=ei,e^d=eh,f^c=fi,f^e=fj,g^c=ghij,g^d=gi,h^d=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306761, #72306775, #72306791, …
maximal normal subgroup of	#72306761, #72306775, #72306791, …

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