#6786401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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

1614 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)(17,18,22,19)(20,27,31,32)(21,24,28,25)(23,29,26,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,31,26)(21,24,28,25)(27,29,32,30);(2,6)(3,8)(10,13)(14,16)(17,20,28,29,22,31,21,30)(18,23,24,32,19,26,25,27);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,28)(18,24,19,25)(20,29,31,30)(23,27,26,32);(10,13)(14,16)(17,21,22,28)(18,24,19,25)(20,30,31,29)(23,32,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fh,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,d^c=di

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, 6693597);