#423620

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1559 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=ei,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, 330816);