#7586754

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
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×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16);(9,11,13,12)(10,16,14,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(18,23)(22,25)(24,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7493950);