#6491430

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62861	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2859 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)(9,10,14,18,15,12,26,19)(11,20,23,21,25,33,24,16)(13,17,29,31,28,27,39,32)(22,34,37,35,36,40,38,30);(1,2)(3,6)(4,5)(7,8)(9,11,15,25)(10,16,12,21)(13,22,28,36)(14,24,26,23)(17,30,27,35)(18,33,19,20)(29,38,39,37)(31,40,32,34);(2,5)(6,8)(9,12)(10,15)(11,21)(13,27)(14,19)(16,25)(17,28)(18,26)(20,24)(22,35)(23,33)(29,32)(30,36)(31,39)(34,38)(37,40);(2,5)(6,8)(9,13)(10,17)(11,22)(12,27)(14,29)(15,28)(16,30)(18,31)(19,32)(20,34)(21,35)(23,37)(24,38)(25,36)(26,39)(33,40);(1,3,4,7)(2,6,5,8)(13,28)(17,27)(22,36)(29,39)(30,35)(31,32)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,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, 6398626);