#6465788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=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, 6372984);