#6637234

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,26,27,32)(20,23,31,30)(21,29,28,25);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(19,27)(20,31)(23,30)(26,32);(11,15)(13,16)(17,19,28,23,22,27,21,30)(18,20,29,32,24,31,25,26);(2,6)(3,8)(11,15)(13,16)(17,20,22,31)(18,23,24,30)(19,29,27,25)(21,26,28,32);(2,6)(5,7)(10,14)(13,16)(18,24)(20,31)(25,29)(26,32)

Polycyclic presentation:

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

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, 6544430);