#6500185

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

3157 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^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, 6407381);