#6833122

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	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1327 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=fi,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, 6740318);