Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1824 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 319 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 — 3 generators on 72 points:
(3,6,7,8)(9,10,14,21,16,19,15,22)(11,23,26,43,28,17,27,42)(12,29,32,62,30,63,33,50)(13,34,37,48,39,51,38,20)(18,44,45,31,47,36,46,64)(24,53,56,72,54,68,57,71)(25,58,60,49,35,66,61,41)(40,67,69,65,52,59,70,55);(1,2,5,4)(3,6,7,8)(9,11)(10,17)(12,24)(13,35)(14,27)(15,26)(16,28)(18,40)(19,23)(20,49)(21,43)(22,42)(25,39)(29,53)(30,54)(31,65)(32,57)(33,56)(34,58)(36,67)(37,60)(38,61)(41,48)(44,59)(45,70)(46,69)(47,52)(50,72)(51,66)(55,64)(62,71)(63,68);(1,3)(2,6)(4,8)(5,7)(9,12,15,32,16,30,14,33)(10,18,22,45,19,47,21,46)(11,24,26,57,28,54,27,56)(13,36,38,64,39,44,37,31)(17,40,42,70,23,52,43,69)(20,50,51,29,48,62,34,63)(25,59,60,65,35,67,61,55)(41,71,58,68,49,72,66,53)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=dg,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42591);