#152817

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 245 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=di,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, 60013);