Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 52 points:
(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,25,31,26,32,40,49,27,29,41,45,24,30,38,47)(23,33,35,50,34,43,51,48,36,46,39,52,37,28,42,44);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,37)(25,39)(26,34)(27,36)(29,43)(30,46)(31,48)(32,33)(35,41)(38,51)(40,42)(44,45)(47,50)(49,52);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,29)(28,43)(30,32)(31,45)(33,46)(44,48)(47,49)(50,52)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,d^b=di,e^b=egi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34131);