#72322973

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₈
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#2184	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ C₂×C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=h,e^2=ij,f^2=i,g^2=i,i^2=j,b^a=bg,d^a=de,d^b=defij,d^c=dh,e^a=ei,e^b=ei,e^d=ei,f^a=fij,f^b=fij,f^d=fij,g^a=gij,g^b=gij,g^d=gij,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322980
maximal normal subgroup of	#72322980

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