Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1548 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 1548 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,27)(22,28)(23,26)(29,37)(30,39)(31,38)(32,40)(33,42)(34,43)(35,41)(36,44)(45,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,30,23,31,21,36)(18,24,28,39,26,38,27,44)(20,32,34,45,35,46,33,29)(25,40,43,47,41,48,42,37);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20,21,33,23,35,22,34)(18,25,27,42,26,41,28,43)(19,29,30,32,31,45,36,46)(24,37,39,40,38,47,44,48);(3,8)(5,6)(11,15)(13,16)(18,26)(24,38)(25,41)(27,28)(37,47)(39,44)(40,48)(42,43)Polycyclic presentation:
2^9|b^2=h,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299225);