Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,17)(8,22,25,13)(9,14)(10,29)(11,16)(12,20)(15,32)(18,46,49,44)(19,45)(21,42)(23,54,55,37)(24,38,31,43)(26,39,57,65)(27,58)(28,41)(30,35)(33,53,36,48)(34,51)(40,60)(47,62,68,67)(50,61,56,64)(52,66)(59,63);(5,7,11,21)(6,12,16,35)(8,18,25,49)(9,19,28,52)(10,30,32,20)(13,36,22,33)(14,34,41,63)(15,42,29,17)(23,47,55,68)(24,53,31,48)(26,56,57,50)(27,59,60,51)(37,64,54,61)(38,46,43,44)(39,62,65,67)(40,66,58,45);(2,4)(5,8)(6,13)(7,18)(9,26)(10,31)(11,25)(12,33)(14,39)(15,43)(16,22)(17,44)(19,50)(20,53)(21,49)(23,27)(24,32)(28,57)(29,38)(30,48)(34,62)(35,36)(37,40)(41,65)(42,46)(45,61)(47,51)(52,56)(54,58)(55,60)(59,68)(63,67)(64,66);(5,9,11,28)(6,14,16,41)(7,19,21,52)(8,23)(10,27,32,60)(12,34,35,63)(13,37)(15,40,29,58)(17,45,42,66)(18,47)(20,51,30,59)(22,54)(24,26)(25,55)(31,57)(33,61)(36,64)(38,39)(43,65)(44,67)(46,62)(48,50)(49,68)(53,56)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288673);