#7624499

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1370 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(18,22)(19,25)(21,28)(23,31)(24,27)(26,32)(29,30);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,18,20,22)(19,23,26,30)(21,24,27,28)(25,29,32,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19,20,26)(18,23,22,30)(21,29,27,31)(24,32,28,25);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)(17,18,20,22)(19,23,26,30)(21,24,27,28)(25,29,32,31)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,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, 7531695);