#302511

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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

693 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)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19)(25,26,30,35)(27,33,37,32)(28,39,38,40)(29,36,31,34);(1,3,5,4)(2,6,8,7)(10,14)(15,22)(16,21)(17,24)(25,27)(26,32)(28,29)(30,37)(31,38)(33,35)(34,39)(36,40);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)(25,28,29,27,30,38,31,37)(26,33,34,39,35,32,36,40);(1,4,5,3)(2,7,8,6)(10,14)(15,22)(16,21)(17,24)(25,29,30,31)(26,34,35,36)(27,28,37,38)(32,39,33,40)

Polycyclic presentation:

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

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, 209707);