#6840652

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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)(11,14)(12,13)(15,16);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,23,27)(19,21,29,31)(20,25,22,32)(24,28,30,26);(10,13)(14,16)(17,19,22,26,23,29,20,28)(18,24,25,21,27,30,32,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,22)(18,25,27,32)(19,28,29,26)(21,30,31,24);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,20,24,23,31,22,30)(18,26,32,19,27,28,25,29)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fgi,e^2=ghi,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, 6747848);