#337813

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 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)(12,17)(13,15)(14,19)(16,21)(20,23)(22,24)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(25,27)(26,31)(28,35)(29,37)(30,34)(32,33)(36,39)(38,40);(1,4,5,8)(2,6,3,7)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,28,30,36)(26,32,29,38)(27,33,34,40)(31,39,37,35);(2,3)(6,7)(10,16)(11,19)(13,21)(14,22)(15,17)(18,23)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=ehi,d^2=i,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, 245009);