#6629175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628867
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4184 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)(9,10,15,20)(11,21,24,33)(12,17,26,32)(13,19,14,18)(16,27,30,23)(22,36,39,35)(25,34,28,31)(29,37,40,38);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,23)(14,27)(15,24)(17,29)(18,21)(19,33)(20,30)(25,37)(26,39)(28,38)(31,35)(32,40)(34,36);(9,12)(10,17)(11,22)(13,25)(14,28)(15,26)(16,29)(18,31)(19,34)(20,32)(21,35)(23,37)(24,39)(27,38)(30,40)(33,36);(9,13,15,14)(10,18,20,19)(11,23,24,27)(12,25,26,28)(16,21,30,33)(17,31,32,34)(22,37,39,38)(29,35,40,36)

Polycyclic presentation:

2^9|a^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,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, 6536371);