#414924

order	512
nilpotency class	3
isoclinism	isoclinic class of #394797
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322120);