Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,12,16,11,18,10,17)(7,19,23,38,24,37,22,13)(8,25,28,14,30,41,29,42)(9,31,34,44,33,43,32,15)(20,39,48,45,26,35,49,54)(21,50,52,62,53,36,51,46)(27,57,58,55,60,40,59,64)(47,66,67,63,56,65,68,61);(5,7)(6,13,18,38)(8,26,30,20)(9,21)(10,22)(11,24)(12,23)(14,39)(15,36,44,50)(16,19,17,37)(25,54)(27,56,60,47)(28,49,29,48)(31,46,43,62)(32,51)(33,53)(34,52)(35,42)(40,63)(41,45)(55,66)(57,61)(58,68,59,67)(64,65);(1,2,3,4)(5,8,12,29,11,30,10,28)(6,14,16,25,18,42,17,41)(7,20,22,48,24,26,23,49)(9,27,34,59,33,60,32,58)(13,35,37,54,38,39,19,45)(15,40,31,55,44,57,43,64)(21,47,51,67,53,56,52,68)(36,61,62,66,50,63,46,65);(5,9,12,34,11,33,10,32)(6,15,17,43,18,44,16,31)(7,21,23,52,24,53,22,51)(8,27,29,59,30,60,28,58)(13,36,37,62,38,50,19,46)(14,40,41,64,42,57,25,55)(20,47,49,68,26,56,48,67)(35,61,54,66,39,63,45,65)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276377);