Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13902 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,19,23,10)(5,24)(6,27)(7,12)(8,13)(9,17)(11,25)(15,42,45,35)(16,32)(18,34)(20,31,44,57)(21,51,54,55)(22,37,30,38)(26,40)(28,33)(29,39)(36,52,49,53)(41,46)(43,62,63,56)(47,58)(48,59)(50,61,64,60);(1,3,7,17,8,18,6,28)(2,9,12,34,13,33,27,14)(4,20,22,45,23,44,30,15)(5,16,25,47,26,48,29,46)(10,35,38,57,19,42,37,31)(11,32,40,58,39,59,24,41)(21,50,53,63,54,64,52,43)(36,60,55,62,49,61,51,56);(1,4)(2,10)(3,15)(5,21)(6,22)(7,30)(8,23)(9,31)(11,36)(12,37)(13,19)(14,35)(16,43)(17,44)(18,45)(20,28)(24,55)(25,52)(26,54)(27,38)(29,53)(32,56)(33,57)(34,42)(39,49)(40,51)(41,60)(46,50)(47,64)(48,63)(58,61)(59,62);(1,5)(2,11)(3,16)(4,21)(6,29)(7,25)(8,26)(9,32)(10,36)(12,40)(13,39)(14,41)(15,43)(17,47)(18,48)(19,49)(20,50)(22,53)(23,54)(24,27)(28,46)(30,52)(31,56)(33,59)(34,58)(35,60)(37,51)(38,55)(42,61)(44,64)(45,63)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268913);