#6500184

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3130 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:

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

Polycyclic presentation:

2^9|a^2=fi,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, 6407380);