#6491493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470231
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	C₂⁴	C₂⁵
prime radical	#62847	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3010 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,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, 6398689);