#337784

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 244980);