#6822373

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	#25109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,26)(19,21,27,24)(20,25,31,32)(23,28,30,29);(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,30)(21,26)(22,27)(23,31)(25,28)(29,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,30,19,22,31,23,27)(18,25,29,21,26,32,28,24);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(18,26)(21,24)(25,32)(28,29);(10,14)(15,16)(17,21,23,25,22,24,30,32)(18,27,28,31,26,19,29,20)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=g,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, 6729569);