#72309121

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#386
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂²×#447	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

4487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=hj,d^2=e,e^2=h,g^2=j,b^a=bg,d^a=df,d^b=dhj,d^c=di,e^a=ej,f^b=fi,f^d=fj,g^a=gj,g^b=gj,g^d=gi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309133, #72309231, #72309257
maximal normal subgroup of	#72309133, #72309231, #72309257

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