Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 306 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18,22,26,21,27,36,44,23,28,38,45,37,25,41,43)(19,20,30,33,29,34,40,47,31,35,42,48,46,32,24,39);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19,22,30,21,29,36,40,23,31,38,42,37,46,41,24)(18,20,26,33,27,34,44,47,28,35,45,48,25,32,43,39);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(17,20)(18,24)(19,26)(21,34)(22,33)(23,35)(25,42)(27,30)(28,40)(29,44)(31,45)(32,37)(36,47)(38,48)(39,41)(43,46)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,e^b=eg,d^c=dg,e^c=egi,f^c=fg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45018);