Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1282 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,18,14,19,30,41,15,20,31,42,32,43,55,65)(11,21,23,44,25,45,48,66,24,37,47,60,49,61,67,71)(12,17,27,38,29,40,53,63,28,39,52,62,54,64,69,72)(16,33,34,46,36,56,58,26,35,50,57,51,59,68,70,22);(2,5,7,6)(9,11)(10,16)(12,26,28,22)(13,23)(14,25)(15,24)(17,37,39,21)(18,34)(19,36)(20,35)(27,50,52,33)(29,51,54,46)(30,48)(31,47)(32,49)(38,60,62,44)(40,61,64,45)(41,58)(42,57)(43,59)(53,68,69,56)(55,67)(63,71,72,66)(65,70)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 610);