#6837467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42658	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,23,27)(19,28,31,22)(20,32,29,25)(21,30,24,26);(9,10)(11,16)(12,15)(13,14)(17,19,23,31)(18,22,27,28)(20,21,29,24)(25,30,32,26);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20,24,19,23,29,21,31)(18,25,26,22,27,32,30,28);(9,12,13,11)(10,15,14,16)(17,21,23,24)(18,26,27,30)(19,29,31,20)(22,25,28,32);(1,4,5,3)(2,6,7,8)(17,22,21,25,23,28,24,32)(18,19,30,20,27,31,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6744663);