#6466477

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 6373673);