#295722

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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

659 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=fh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^b=gi,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, 202918);