#303000

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=fh,d^2=ei,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, 210196);