#6466432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,29,32,31);(2,5)(6,8)(17,19,22,26)(18,24,23,21)(20,29,30,31)(25,32,28,27);(9,11,12,16)(10,13,14,15)(17,20,18,25,22,30,23,28)(19,27,21,31,26,32,24,29);(17,21,23,19,22,24,18,26)(20,29,28,32,30,31,25,27);(1,3,4,7)(2,6,5,8)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,29,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373628);