#6696855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#23869	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1702 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6604051);