#290954

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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fhi,d^2=hi,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, 198150);