#6500181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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	#62886	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3074 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=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, 6407377);