#7624709

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1356 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531905);