#6432084

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62686	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3365 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,23)(17,28)(19,36)(20,30)(21,34)(24,35)(26,31)(27,33)(29,38)(32,39)(37,40);(2,5)(6,8)(9,11)(10,15)(12,19)(13,26)(14,20)(16,32)(17,34)(18,30)(21,28)(22,36)(23,39)(24,37)(25,31)(27,38)(29,33)(35,40);(9,12,14,24)(10,16,18,33)(11,19,20,37)(13,27,28,23)(15,29,30,32)(17,35,25,22)(21,39,26,38)(31,40,34,36);(1,3,4,7)(2,6,5,8);(11,20)(15,30)(19,37)(21,26)(29,32)(31,34)(36,40)(38,39)

Polycyclic presentation:

2^9|c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,e^b=ei,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339280);