Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133847 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
1408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20,23,32)(12,17)(13,18,27,16)(14,19)(15,21,30,24)(22,36)(25,33,40,31)(26,34)(28,37)(29,39)(35,38);(2,5)(3,7)(9,11,14,23)(10,15,19,30)(12,24,26,21)(13,22,27,38)(16,28,18,29)(17,32,34,20)(25,39,40,37)(31,36,33,35);(10,16)(13,27)(15,28)(17,31)(18,19)(20,35)(22,38)(25,40)(29,30)(32,36)(33,34)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fgi,e^c=ei,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72318004, #72318204, #72318719, … |
| maximal normal subgroup of | #72318004, #72318204, #72318719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41251);