#457210

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1345 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364406);