#6491819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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:

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

Polycyclic presentation:

2^9|b^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6399015);