Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144372 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
647 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,6,10,13)(4,7,11,15)(9,12,16,14)(18,22)(19,25)(20,28)(23,31)(24,26)(29,32);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(18,23)(20,28)(22,31)(24,29)(26,32)(27,30);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,f^b=fi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72268087, #72268088, #72268326, … |
| maximal normal subgroup of | #72268087, #72268088, #72268326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51618);