#394120

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3864 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)(9,10,16,21)(11,22,25,30)(12,18,29,36)(13,24,17,31)(14,20,15,19)(23,38,40,34)(26,27,39,33)(28,37,32,35);(2,4)(9,11)(10,17)(12,26)(13,21)(14,31)(15,24)(16,25)(18,34)(19,30)(20,22)(23,28)(27,37)(29,39)(32,40)(33,35)(36,38);(6,8)(9,12)(10,18)(11,23)(13,27)(14,32)(15,28)(16,29)(17,33)(19,37)(20,35)(21,36)(22,38)(24,39)(25,40)(26,31)(30,34);(9,13,16,17)(10,11,21,25)(12,27,29,33)(14,30,15,22)(18,23,36,40)(19,24,20,31)(26,37,39,35)(28,38,32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301316);