#297165

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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

631 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)(9,10,14,18)(11,19,22,16)(12,15,17,13)(20,24,23,21)(28,36)(30,38)(32,37)(35,40);(2,5)(6,7)(9,11,14,22)(10,16,18,19)(12,23,17,20)(13,21,15,24)(25,26)(27,33)(28,37)(29,31)(30,35)(32,36)(34,39)(38,40);(9,12,15,18,14,17,13,10)(11,20,21,19,22,23,24,16)(25,27)(26,30,31,38)(28,35)(29,34)(32,33,37,39)(36,40);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,17,18,12)(11,21,22,24)(16,20,19,23)(27,34)(30,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=f,d^2=e,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=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204361);