#290902

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

712 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,16,22,19)(12,13,17,14)(20,24,21,23)(25,26,31,34)(27,32,37,38)(28,36,39,40)(29,33,30,35);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,20)(14,21)(15,22)(17,24)(18,19)(25,27)(26,32)(28,35)(29,36)(30,40)(31,37)(33,39)(34,38);(9,12,14,10,15,17,13,18)(11,19,21,23,22,16,20,24)(25,28,29,32,31,39,30,38)(26,27,35,40,34,37,33,36);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,20,22,21)(16,24,19,23)(25,29,31,30)(26,33,34,35)(27,36,37,40)(28,32,39,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fi,d^2=h,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, 198098);