Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
800 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,10,7,12,5,11)(3,13,9,8,15,16,14,4)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,30)(25,35)(29,39)(31,40)(32,43)(34,41)(36,42)(37,44)(45,46)(47,48);(1,3,7,15)(2,8,12,4)(5,14,6,9)(10,16,11,13)(17,19,23,32)(18,24,27,40)(20,34,37,46)(21,31,22,30)(25,41,42,45)(26,43,38,28)(29,35,47,36)(33,39,44,48);(1,4,7,8)(2,9,12,14)(3,10,15,11)(5,16,6,13)(17,20,23,37)(18,25,27,42)(19,29,32,47)(21,36,22,35)(24,39,40,48)(26,44,38,33)(28,45,43,41)(30,34,31,46)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18674);