#335600

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 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)(3,7,8,5)(9,10)(11,18)(12,14)(13,15)(16,22)(17,21)(19,24)(20,23)(26,30)(27,33)(32,37)(35,38);(1,3,4,8)(2,5,6,7)(10,14)(15,17)(16,23)(18,24)(25,26)(27,34)(28,30)(29,37)(31,35)(32,36)(33,40)(38,39);(10,15)(13,21)(14,17)(16,24)(18,23)(20,22)(25,27,28,33)(26,31,30,39)(29,38,36,35)(32,40,37,34);(2,6)(3,8)(9,11,12,19)(10,16,14,23)(13,22,21,20)(15,24,17,18)(26,30)(31,39)(32,37)(34,40)

Polycyclic presentation:

2^9|a^2=ehi,b^2=ehi,c^2=i,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=deh,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, 242796);