#303042

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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)(9,10,15,19)(11,16,17,23)(12,22,24,21)(13,18,14,20)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,18)(13,21)(14,22)(15,17)(19,23)(20,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,34)(38,40);(9,12,14,16,15,24,13,23)(10,17,18,22,19,11,20,21)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,18,19,20)(11,21,17,22)(12,23,24,16)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=ehi,b^2=ei,c^2=f,d^2=eh,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=di,g^c=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, 210238);