#6466687

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 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)(12,16)(13,15);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,26)(20,27)(21,24)(22,25)(23,28)(29,30)(31,32);(9,12,13,11)(10,14,15,16)(17,19,26,18,22,24,21,25)(20,27,32,23,29,30,28,31);(17,20,21,28,22,29,26,32)(18,23,19,27,25,31,24,30);(1,3,4,7)(2,6,5,8)(17,21,22,26)(18,24,25,19)(20,28,29,32)(23,30,31,27)

Polycyclic presentation:

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

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, 6373883);