#6817897

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6725093);