Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,21,14,20,37,22)(11,23,28,45,26,17,40,44)(12,29,32,47,31,64,33,66)(13,19,36,51,35,50,15,38)(18,46,49,65,48,39,52,30)(24,54,58,68,56,63,59,70)(25,43,27,61,34,42,60,67)(41,62,69,57,53,72,71,55);(2,5)(6,8)(9,11)(10,17)(12,24)(13,34)(14,26)(15,27)(16,28)(18,41)(19,43)(20,23)(21,44)(22,45)(25,35)(29,63)(30,57)(31,56)(32,58)(33,59)(36,60)(37,40)(38,67)(39,62)(42,50)(46,72)(47,70)(48,53)(49,69)(51,61)(52,71)(54,64)(55,65)(66,68);(1,3,4,7)(2,6,5,8)(9,12,14,31)(10,18,20,48)(11,24,26,56)(13,30,35,65)(15,39,36,46)(16,32,37,33)(17,41,23,53)(19,47,50,66)(21,49,22,52)(25,55,34,57)(27,62,60,72)(28,58,40,59)(29,51,64,38)(42,68,43,70)(44,69,45,71)(54,67,63,61);(9,13,16,36,14,35,37,15)(10,19,21,51,20,50,22,38)(11,25,28,27,26,34,40,60)(12,30,32,46,31,65,33,39)(17,42,44,67,23,43,45,61)(18,47,49,64,48,66,52,29)(24,55,58,62,56,57,59,72)(41,68,69,63,53,70,71,54)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=g,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403134);