#6814157

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	#42658	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,30,24)(20,32,23,25)(21,26,29,31);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,29)(20,25)(21,30)(22,27)(23,32)(24,31)(26,28);(10,14)(13,16)(17,19,20,26,22,30,23,31)(18,24,32,21,27,28,25,29);(1,4,6,5)(2,7,3,8)(17,20,22,23)(18,25,27,32)(19,26,30,31)(21,24,29,28);(1,5,6,4)(2,7,3,8)(17,21,23,24,22,29,20,28)(18,26,25,19,27,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6721353);