#345636

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252832);