#331580

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=eg,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238776);