#305011

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=ef,d^2=eh,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, 212207);