Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,22,26,23,27,21,35)(19,24,31,39,29,38,30,45)(20,32,33,25,34,40,36,41)(28,42,44,37,43,48,46,47);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,42)(33,44)(34,43)(35,45)(36,46)(40,48)(41,47);(10,13)(14,16)(17,20,22,33,23,34,21,36)(18,25,26,40,27,41,35,32)(19,28,31,44,29,43,30,46)(24,37,39,48,38,47,45,42);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,43)(30,31)(37,47)(39,45)(42,48)(44,46)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=h,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282900);