#6629976

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1806 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)(11,13)(12,14)(15,16)(17,18,22,19)(20,28,30,32)(21,25,23,27)(24,31,26,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,26,30,24)(21,25,23,27)(28,29,32,31);(10,14)(13,16)(18,19)(24,26)(25,27)(28,32);(17,20,22,30)(18,24,19,26)(21,29,23,31)(25,32,27,28);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537172);