Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121833 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1445 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
530 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 530 conjugacy classes of subgroups — too many to draw. Only its 114 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)(4,7,12,14)(5,8)(6,11)(10,13,16,15)(17,18)(19,27,30,23)(20,31)(21,34)(22,25)(24,33)(26,36)(28,43,44,48)(29,38,47,42)(32,41)(35,40)(37,45,39,46);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,21,29,22,30,36,47)(18,23,34,42,25,27,26,38)(20,28,35,46,32,44,33,45)(24,39,31,43,40,37,41,48);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,20,21,35,22,32,36,33)(18,24,34,31,25,40,26,41)(19,28,29,46,30,44,47,45)(23,37,42,48,27,39,38,43)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=ei,g^b=gh,d^c=dhi,e^c=ehi,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29029);