Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #525 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,23,27,21,29,39,32,24,28,42,47,22,26,31,46)(19,30,25,43,33,20,35,37,34,36,44,48,40,38,45,41);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,31,45,22,40,42,44,24,34,39,35,21,33,23,25)(18,20,32,48,26,30,27,37,28,38,46,43,29,36,47,41);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,25)(19,32)(21,36)(22,30)(23,41)(24,38)(26,45)(27,40)(28,44)(29,35)(31,48)(33,47)(34,46)(37,42)(39,43)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,e^b=eg,d^c=dg,e^c=ei,f^c=fg74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44040);