#6843790

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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:

(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,25)(19,27,28,21)(20,29,31,24)(23,26,30,32);(1,2)(3,7)(4,8)(5,6)(17,19)(18,21)(20,30)(22,28)(23,31)(24,32)(25,27)(26,29);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,30,19,22,31,23,28)(18,24,32,21,25,29,26,27);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,25)(21,27)(24,29)(26,32);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(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=fh,d^2=gi,e^2=fgh,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, 6750986);