#6629374

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	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4031 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)(9,10,15,19)(11,16,23,31)(12,24,26,32)(13,18,14,27)(17,28,33,25)(20,34,36,39)(21,30,22,37)(29,38,40,35);(2,5)(6,8)(9,11,15,23)(10,16,19,31)(12,20,26,36)(13,21,14,22)(17,29,33,40)(18,30,27,37)(24,34,32,39)(25,35,28,38);(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,32)(19,33)(21,35)(22,38)(23,36)(24,27)(30,39)(31,40)(34,37);(13,14)(18,27)(21,22)(24,32)(25,28)(30,37)(34,39)(35,38);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,31,37)(17,32,33,24)(20,35,36,38)(29,39,40,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,h^c=hi,e^d=ei,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, 6536570);