Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1120 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3,4)(9,12)(13,14)(15,24)(16,29)(17,34)(18,22)(19,23)(20,26)(21,32)(25,42)(27,37)(28,36)(30,41)(31,40)(33,39)(35,43)(38,44);(5,6)(7,9)(8,10)(11,12)(13,15,19,28)(14,20,23,37)(16,30,33,44)(17,27,18,26)(21,38,40,41)(22,36,34,24)(25,32,43,31)(29,35,39,42);(5,7)(6,9)(8,11)(10,12)(13,16,19,33)(14,21,23,40)(15,25,28,43)(17,31,18,32)(20,35,37,42)(22,39,34,29)(24,41,36,38)(26,30,27,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,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, 18669);