#6686100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6593296);