#339222

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=ehi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246418);