#6833794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6740990);