#6629598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628847
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1911 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,31)(22,27)(23,24)(26,29)(30,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,24,27,28)(20,30,21,29)(25,26,31,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,21)(18,25,27,31)(19,29,23,30)(24,32,28,26);(17,21,22,20)(18,26,27,32)(19,29,23,30)(24,31,28,25);(18,27)(24,28)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536794);