#6629172

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	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4205 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)(11,15)(12,22)(13,25)(14,17)(16,23)(18,27)(19,33)(20,36)(21,29)(24,32)(26,31)(28,34)(30,38)(35,40)(37,39);(2,5)(6,8)(9,11,14,21)(10,15,17,29)(12,19,24,35)(13,20,27,38)(16,28,31,39)(18,30,25,36)(22,33,32,40)(23,34,26,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,31)(18,32)(20,37)(21,35)(22,25)(23,27)(29,39)(30,40)(33,36)(34,38);(13,27)(18,25)(20,38)(22,32)(23,26)(30,36)(33,40)(34,37);(10,17)(15,29)(16,31)(18,25)(22,32)(28,39)(30,36)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=f,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, 6536368);