#152818

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

227 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=d,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60014);