#335214

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×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 650 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)(9,10,14,19)(11,15,12,21)(13,17,23,18)(16,20,24,22)(25,26)(27,35)(28,33)(29,31)(30,34)(32,37)(36,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11,14,12)(10,15,19,21)(13,20,23,22)(16,17,24,18)(26,31)(27,36)(28,30)(32,39)(35,40)(37,38);(9,12,14,11)(10,16,19,24)(13,20,23,22)(15,17,21,18)(25,27,29,37)(26,32,31,35)(28,36,30,38)(33,40,34,39);(2,6)(3,8)(10,17)(13,23)(15,24)(16,21)(18,19)(20,22)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=ehi,b^2=ehi,c^2=hi,d^2=i,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, 242410);