Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121843 |
| 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 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
507 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 507 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)(4,7,12,14)(5,8)(6,11)(10,13,16,15)(17,18,23,26,22,27,21,36)(19,28,32,39,31,40,30,24)(20,25,34,43,35,44,37,42)(29,41,45,48,46,47,33,38);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,23,32,22,31,21,30)(18,24,36,40,27,39,26,28)(20,33,34,29,35,45,37,46)(25,41,42,38,44,47,43,48);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,20,23,34,22,35,21,37)(18,25,36,42,27,44,26,43)(19,29,32,45,31,46,30,33)(24,38,40,47,39,48,28,41)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=dhi,e^c=ehi,f^c=fh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29040);