#72239724

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#379
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#7711145	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

4875 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:

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

Polycyclic presentation:

2^10|a^2=hj,b^2=e,c^2=egi,d^2=ij,e^2=j,f^2=h,g^2=j,b^a=bf,c^a=cfhij,c^b=cg,d^a=dij,d^b=di,d^c=di,e^a=ei,e^c=ei,f^a=fh,f^b=fhi,f^c=fi,g^a=gh,g^b=gij,g^c=gi

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239752
maximal normal subgroup of	#72239752

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