#6786351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(3,7)(4,6)(5,8)(17,18)(19,26)(20,21)(22,25)(23,30)(24,29)(27,28)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,25,26)(20,27,30,32)(21,31,23,28);(2,6)(3,8)(10,13)(14,16)(17,20,21,18,22,30,23,25)(19,27,31,24,29,32,28,26);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,27)(21,31)(22,29)(23,28)(25,26)(30,32);(10,13)(14,16)(17,21,22,23)(18,20,25,30)(19,28,29,31)(24,32,26,27)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6693547);