#414865

order	512
nilpotency class	3
isoclinism	isoclinic class of #394583
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 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)(11,21,24,36)(12,25,29,35)(13,20,14,19)(16,30,32,23)(17,27,34,28)(22,33,26,31)(37,40,39,38);(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,33)(18,32)(19,36)(20,21)(22,29)(25,39)(27,38)(28,40)(31,34)(35,37);(9,12,13,27,15,29,14,28)(10,17,19,35,18,34,20,25)(11,22,23,38,24,26,30,40)(16,31,21,37,32,33,36,39);(1,3,4,8)(2,5,6,7)(10,18)(16,32)(17,34)(19,20)(21,36)(25,35)(31,33)(37,39)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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, 322061);