Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,18,12,17,10,16)(7,19,22,37,24,46,23,13)(8,25,29,40,30,14,28,41)(9,15,33,45,34,44,32,43)(20,38,50,62,51,47,26,35)(21,48,53,64,55,66,54,36)(27,56,58,42,59,39,31,60)(49,63,52,65,68,67,57,61);(1,2)(3,4)(5,7)(6,13,17,37)(8,26)(9,21)(10,22)(11,23)(12,24)(14,38,25,47)(15,36,44,64)(16,19,18,46)(20,28)(27,57)(29,51)(30,50)(31,49)(32,53)(33,54)(34,55)(35,40,62,41)(39,63,56,67)(42,65,60,61)(43,48,45,66)(52,59)(58,68);(2,4)(5,8,12,30)(6,14,17,25)(7,20,24,51)(9,31,34,58)(10,29,11,28)(13,35,37,62)(15,42,44,60)(16,41,18,40)(19,47,46,38)(21,52,55,57)(22,26,23,50)(27,33,59,32)(36,63,64,67)(39,45,56,43)(48,61,66,65)(49,54,68,53);(5,9,11,33,12,34,10,32)(6,15,18,45,17,44,16,43)(7,21,23,54,24,55,22,53)(8,27,29,58,30,59,28,31)(13,36,46,66,37,64,19,48)(14,39,41,60,25,56,40,42)(20,49,26,57,51,68,50,52)(35,61,47,67,62,65,38,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,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, 287387);