Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6)(7,18)(8,23,28,41)(9,14,32,46)(10,16)(11,17)(12,21)(13,27,42,26)(15,34)(19,49,52,35)(20,48,54,43)(22,37)(24,39,51,47)(25,57)(29,62,53,36)(30,45,31,44)(33,38)(40,59)(50,63)(55,61)(56,67)(58,65)(60,66)(64,68);(2,4)(5,7,11,22)(6,12,17,38)(8,24,28,51)(9,29,32,53)(10,33,34,21)(13,39,42,47)(14,43,46,48)(15,37,16,18)(19,26,52,27)(20,31,54,30)(23,49,41,35)(25,58,61,50)(36,45,62,44)(40,65,66,63)(55,67,57,64)(56,60,68,59);(1,2,3,4)(5,8,11,28)(6,13,17,42)(7,19,22,52)(9,25,32,61)(10,26,34,27)(12,35,38,49)(14,40,46,66)(15,23,16,41)(18,47,37,39)(20,50,54,58)(21,51,33,24)(29,56,53,68)(30,59,31,60)(36,63,62,65)(43,64,48,67)(44,57,45,55);(5,9)(6,14)(7,20)(8,25)(10,31)(11,32)(12,36)(13,40)(15,45)(16,44)(17,46)(18,48)(19,50)(21,53)(22,54)(23,55)(24,56)(26,60)(27,59)(28,61)(29,33)(30,34)(35,63)(37,43)(38,62)(39,64)(41,57)(42,66)(47,67)(49,65)(51,68)(52,58)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=egh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297620);