#6435594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433077
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×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,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, 6342790);