#6466883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,5)(3,7,8,6)(9,10,15,19)(11,12,24,28)(13,17,30,33)(14,20,21,18)(16,25,27,23)(22,26,37,39)(29,40,38,32)(31,35,36,34);(2,5)(6,7)(9,11,21,25,15,24,14,23)(10,16,20,28,19,27,18,12)(13,29,36,37,30,38,31,22)(17,26,35,32,33,39,34,40);(1,3,4,8)(2,6,5,7)(9,12,21,16,15,28,14,27)(10,11,18,23,19,24,20,25)(13,26,36,32,30,39,31,40)(17,22,34,38,33,37,35,29);(9,13,14,31,15,30,21,36)(10,17,20,35,19,33,18,34)(11,22,25,29,24,37,23,38)(12,26,27,40,28,39,16,32);(13,30)(17,33)(22,37)(26,39)(29,38)(31,36)(32,40)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374079);