#457234

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×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=e,e^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,g^b=gi,d^c=de,e^c=ei,h^c=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, 364430);