#6817237

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814819
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=fi,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724433);