Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1502 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8577 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,15,17,12)(4,19)(5,11)(6,22,18,9)(7,13)(8,14)(10,16,26,30)(20,29)(21,32)(23,27,31,25)(24,28);(1,3,7,18,8,17,5,6)(2,9,11,15,14,22,13,12)(4,16,24,31,21,30,20,23)(10,19,25,29,26,32,27,28);(1,4)(2,10)(3,16)(5,20)(6,23)(7,24)(8,21)(9,19)(11,25)(12,28)(13,27)(14,26)(15,29)(17,30)(18,31)(22,32)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,f^b=fi,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219577, #72219578, #72219579, … |
| maximal normal subgroup of | #72219577, #72219578, #72219579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41716);