#6818219

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,20,26,27)(21,29,31,23)(25,30,32,28);(2,5)(3,7)(17,19,22,26)(18,20,24,27)(21,30,31,28)(23,25,29,32);(9,11,12,16)(10,13,14,15)(17,20,18,19,22,27,24,26)(21,30,23,32,31,28,29,25);(2,5)(6,8)(21,31)(23,29)(25,32)(28,30);(1,3,4,7)(2,6,5,8)(17,21,18,23,22,31,24,29)(19,25,27,30,26,32,20,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6725415);