#7033538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313531, #72313635, #72313947
maximal normal subgroup of	#72313531, #72313635, #72313947

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6940734);