#6812098

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	#42463	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,29,24)(20,25,30,21)(23,27,32,31);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,24,26,28)(20,27,30,31)(21,32,25,23);(10,14)(13,16)(17,20,23,28,22,30,32,24)(18,25,27,29,26,21,31,19);(2,6)(5,7)(18,26)(20,30)(24,28)(27,31);(3,8)(5,7)(17,21,32,29,22,25,23,19)(18,20,31,24,26,30,27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,e^2=hi,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, 6719294);