#72240770

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂²×#493	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13759 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=f,d^2=h,e^2=i,i^2=j,c^a=cg,c^b=cg,d^a=dgh,d^b=dj,d^c=dgj,e^a=ei,e^b=ej,e^c=egh,e^d=ej,i^a=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240787, #72265709, #72308733
maximal normal subgroup of	#72240787, #72265709, #72308733

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