Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1442 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
514 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,14)(10,15,13,16)(17,18,22,27,23,28,21,26)(19,29,32,24,33,40,31,41)(20,34,36,25,38,43,37,44)(30,46,47,42,48,39,35,45);(1,3)(2,6)(4,13,14,10)(5,11)(7,16,12,15)(8,9)(17,19,21,32,23,33,22,31)(18,24,26,40,28,41,27,29)(20,35,37,30,38,47,36,48)(25,42,34,39,44,45,43,46);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(17,20,21,36,23,38,22,37)(18,25,26,43,28,44,27,34)(19,30,31,47,33,48,32,35)(24,39,29,45,41,46,40,42)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=ehi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28126);