#259365

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

744 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)(3,7)(4,5)(6,8)(9,10,13,17)(11,18,21,15)(12,22,14,16)(19,24,20,23)(25,26,31,27)(28,36,39,40)(29,34,30,33)(32,38,35,37);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,18)(22,24)(25,27,31,26)(28,36,39,40)(29,33,30,34)(32,38,35,37);(11,19)(12,14)(15,23)(16,22)(18,24)(20,21)(25,28,31,39)(26,32,27,35)(29,38,30,37)(33,40,34,36);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,16,17,22)(11,20,21,19)(15,23,18,24)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=e,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=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166561);