#6832538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=fhi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739734);