#251446

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

786 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 138 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)(12,21)(13,18)(14,19)(15,20)(17,23)(22,24)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,38,39,35);(1,3,4,8)(2,5,6,7)(9,11,15,21)(10,16,20,12)(13,17,14,24)(18,23,19,22)(26,30)(29,36)(33,40)(35,38);(9,12,15,16)(10,17,20,24)(11,18,21,19)(13,22,14,23)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,17,21,24)(12,22,16,23)(26,30)(31,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=h,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,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158642);