#349438

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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)(11,15)(12,21)(13,17)(14,18)(16,20)(19,24)(22,23)(25,26,31,29)(27,32,30,35)(28,36,39,40)(33,37,38,34);(1,3,4,8)(2,5,6,7)(9,11,14,13)(10,15,18,17)(12,20,22,19)(16,23,24,21)(25,27,31,30)(26,32,29,35)(28,37,39,34)(33,36,38,40);(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,23)(18,24)(25,28)(26,33)(27,34)(29,38)(30,37)(31,39)(32,40)(35,36);(9,12)(10,16)(11,20)(13,19)(14,22)(15,23)(17,21)(18,24)(25,29,31,26)(27,35,30,32)(28,38,39,33)(34,36,37,40)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

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, 256634);