#336809

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

549 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=hi,c^2=e,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, 244005);