#6826452

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 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)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,31)(21,28)(23,27)(24,29)(25,30)(26,32);(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,22)(20,32,30,21)(25,26,31,28);(2,5)(6,8)(11,16)(13,15)(17,20,21,19,23,30,32,29)(18,25,28,24,27,31,26,22);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,32)(18,26,27,28)(19,20,29,30)(22,25,24,31);(17,22,21,25,23,24,32,31)(18,19,28,30,27,29,26,20)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=g,e^2=fhi,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, 6733648);