#266581

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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

652 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)(11,16)(12,21)(13,18)(14,19)(15,20)(17,23)(22,24)(25,26)(27,31)(28,36)(29,38)(30,34)(32,37)(33,39)(35,40);(1,3,5,4)(2,6,8,7)(9,11,15,21)(10,16,20,12)(13,17,14,24)(18,23,19,22)(25,27,30,28)(26,31,34,36)(29,37,39,35)(32,33,40,38);(9,12,15,16)(10,17,20,24)(11,18,21,19)(13,22,14,23)(25,28,30,27)(26,32,34,40)(29,35,39,37)(31,38,36,33);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,18,20,19)(11,17,21,24)(12,22,16,23)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=h,d^2=ehi,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,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 173777);