#152159

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=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, 59355);