#290875

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 140 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)(11,20)(12,23)(13,19)(14,18)(15,17)(16,22)(21,24)(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)(10,16)(12,15)(13,21)(14,22)(17,24)(18,20)(19,23)(26,30)(27,34)(29,36)(31,38);(9,12,15,22,14,21,13,11)(10,16,19,23,18,20,17,24)(25,27,29,35,28,34,36,39)(26,31,32,33,30,38,37,40);(2,6)(5,7)(9,13,14,15)(10,17,18,19)(11,21,22,12)(16,24,20,23)(26,30)(31,38)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,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,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198071);