Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #380 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
522 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 522 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:
(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,33)(24,38,40,45)(25,31)(26,30)(27,32)(28,35)(29,39,46,41)(34,42,49,52)(36,44)(37,43)(47,48,51,50);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,27,37,25,35)(22,28,31,43,32,44,30,33)(24,34,41,50,40,49,39,48)(29,42,38,51,46,52,45,47);(7,13)(10,16)(14,19)(17,20)(21,24,25,39,27,40,26,41)(22,29,30,45,32,46,31,38)(23,34,35,48,37,49,36,50)(28,42,33,47,44,52,43,51)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15689);