#335155

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 662 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242351);