#6465789

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=hi,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, 6372985);