#414773

order	512
nilpotency class	3
isoclinism	isoclinic class of #394283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1386 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)(12,24,28,36)(13,29,14,19)(16,30,32,22)(17,26,35,27)(21,31,38,39)(25,34,37,40);(9,11,13,22,15,23,14,30)(10,16,19,20,18,32,29,33)(12,25,26,21,28,37,27,38)(17,34,36,39,35,40,24,31);(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,31)(18,35)(19,36)(20,34)(22,37)(23,38)(24,29)(25,30)(32,39)(33,40);(1,3,4,8)(2,5,6,7)(10,18)(16,32)(17,35)(19,29)(20,33)(24,36)(31,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 321969);