#6817890

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1259 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)(17,18,21,23)(19,20,24,30)(22,25,27,28)(26,31,32,29);(1,2)(3,6)(4,5)(7,8)(18,23)(19,26)(20,29)(22,27)(24,32)(30,31);(9,11,12,16)(10,13,14,15)(17,19,27,32,21,24,22,26)(18,20,28,29,23,30,25,31);(1,3,4,7)(2,6,5,8)(18,23)(20,30)(25,28)(29,31);(2,5)(6,8)(17,20,22,31,21,30,27,29)(18,24,25,32,23,19,28,26)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=fi,e^2=h,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, 6725086);