#418463

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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1682 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)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,29)(22,31)(24,33)(25,32)(27,40)(28,36)(30,35)(34,37)(38,39);(9,11,15,25)(10,16,20,33)(12,27,30,37)(13,24,14,23)(17,34,36,40)(18,32,19,21)(22,29,38,28)(26,39,35,31);(2,4)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,34)(23,33)(25,32)(26,29)(27,39)(28,35)(30,36)(31,37)(38,40);(6,8)(9,12,14,29,15,30,13,28)(10,17,19,26,20,36,18,35)(11,22,24,37,25,38,23,27)(16,31,32,40,33,39,21,34)

Polycyclic presentation:

2^9|b^2=i,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, 325659);