#6432654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=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, 6339850);