#6513024

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500015
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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3153 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)(12,23,25,16)(13,17)(14,18)(15,22)(20,31,24,29)(21,33)(26,36,38,32)(27,30)(28,34)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,24)(13,27)(14,22)(16,31)(17,33)(18,19)(20,25)(21,28)(23,29)(26,35)(30,34)(32,39)(36,40)(37,38);(2,5)(3,7)(9,12)(10,16)(11,20)(13,26)(14,25)(15,29)(17,32)(18,23)(19,31)(21,35)(22,24)(27,37)(28,38)(30,39)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(12,25)(13,28)(16,23)(17,34)(20,24)(21,27)(29,31)(30,33);(2,5)(6,8)(9,13,14,28)(10,17,18,34)(11,21,22,27)(12,26,25,38)(15,30,19,33)(16,32,23,36)(20,35,24,37)(29,39,31,40)

Polycyclic presentation:

2^9|a^2=fi,d^2=hi,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,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, 6420220);