#6629179

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
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4034 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,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, 6536375);