#419883

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1624 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 327079);