#6432793

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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)(11,14)(12,13)(15,16)(18,24)(25,32)(26,28)(27,31);(2,5)(6,8)(17,18,22,24)(19,28,20,26)(21,27,23,31)(25,30,32,29);(10,13)(14,16)(17,19,22,20)(18,25,24,32)(21,29,23,30)(26,31,28,27);(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,26,24,28)(21,29,23,30)(25,31,32,27);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,24,31)(19,29,20,30)(25,26,32,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,h^b=hi,h^c=hi,e^d=ei

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, 6339989);