#303001

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

691 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)(3,7,4,6)(9,10,13,16)(11,17,19,15)(12,14,22,23)(18,24,21,20)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,15,16,17)(12,21,22,18)(14,24,23,20)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,37)(39,40);(9,12,14,16,13,22,23,10)(11,18,20,17,19,21,24,15)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,4,5,3)(2,7,8,6)(10,16)(12,22)(15,17)(18,21)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

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, 210197);