#6840650

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

1524 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)(17,18,22,25)(19,27,28,21)(20,29,31,24)(23,26,30,32);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19)(18,21)(20,30)(22,28)(23,31)(24,32)(25,27)(26,29);(10,13)(14,16)(17,20,30,19,22,31,23,28)(18,24,32,21,25,29,26,27);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(21,27)(24,29)(26,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,24,22,27,30,29)(18,19,26,20,25,28,32,31)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fg,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, 6747846);