#7536090

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,28,32,26);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(10,14)(13,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(2,5)(6,8)(10,14)(13,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=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, 7443286);