#6822375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#25108	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(17,18,21,23)(19,25,28,20)(22,24,27,30)(26,29,32,31);(9,10)(11,15)(12,16)(13,14)(18,23)(19,26)(20,29)(22,27)(25,31)(28,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,27,32,21,28,22,26)(18,20,30,29,23,25,24,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(18,23)(20,25)(24,30)(29,31);(10,14)(15,16)(17,20,22,31,21,25,27,29)(18,19,24,26,23,28,30,32)

Polycyclic presentation:

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

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, 6729571);