#6513004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
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×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3057 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)(11,15)(12,22,24,36)(13,18)(14,17)(16,25,33,27)(19,37,23,40)(20,30)(21,31)(26,35)(28,34)(29,38,32,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,23)(13,26)(14,20)(16,32)(17,30)(18,35)(19,24)(21,28)(22,37)(25,38)(27,39)(29,33)(31,34)(36,40);(2,5)(3,7)(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,33)(18,36)(20,23)(21,39)(22,34)(25,28)(26,38)(30,32)(31,40)(35,37);(1,3,4,7)(2,6,5,8)(10,17)(12,24)(15,30)(18,34)(19,23)(25,27)(31,35)(38,39);(2,5)(6,8)(10,17)(11,20)(16,33)(18,34)(19,23)(21,26)(22,36)(38,39)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi

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, 6420200);