#6813605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6720801);