#6629384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536580);