Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| 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 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,27)(10,30)(11,16)(12,20)(13,26)(14,28)(15,34)(18,46)(19,36)(21,41)(23,44)(24,56)(25,39)(29,43)(31,37)(32,42)(33,40)(35,48)(38,59)(45,49)(47,60)(50,62)(51,53)(52,61)(54,57)(55,68)(58,63)(64,67)(65,66);(5,7,11,21)(6,12,16,37)(8,23,26,18)(9,19,29,50)(10,31,34,20)(13,35,22,53)(14,36,40,62)(15,41,30,17)(24,55,58,47)(25,48,32,51)(27,45,43,61)(28,52,33,49)(38,60,54,68)(39,44,42,46)(56,64,63,65)(57,66,59,67);(5,8,10,32,11,26,34,25)(6,13,15,42,16,22,30,39)(7,18,20,51,21,23,31,48)(9,24,33,57,29,58,28,59)(12,35,17,44,37,53,41,46)(14,38,43,63,40,54,27,56)(19,47,52,66,50,55,49,67)(36,60,45,64,62,68,61,65);(1,3,4,2)(5,9,11,29)(6,14,16,40)(7,19,21,50)(8,24,26,58)(10,33,34,28)(12,36,37,62)(13,38,22,54)(15,43,30,27)(17,45,41,61)(18,47,23,55)(20,52,31,49)(25,59,32,57)(35,60,53,68)(39,56,42,63)(44,64,46,65)(48,67,51,66)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284126);