#260906

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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

644 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)(11,16)(12,21)(13,18)(14,19)(15,20)(17,23)(22,24)(25,26,29,33)(27,35,38,31)(28,34,30,32)(36,39,37,40);(2,5)(6,7)(9,11,15,21)(10,16,20,12)(13,17,14,24)(18,23,19,22)(25,27)(26,31)(28,37)(29,38)(30,36)(32,40)(33,35)(34,39);(9,12,15,16)(10,17,20,24)(11,18,21,19)(13,22,14,23)(27,36)(28,30)(31,39)(32,34)(35,40)(37,38);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,18,20,19)(11,17,21,24)(12,22,16,23)(25,28,29,30)(26,32,33,34)(27,37,38,36)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168102);