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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13920 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 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)(3,9,17,32)(4,19,22,35)(5,11,25,38)(6,13)(7,14)(8,12)(10,23,36,24)(15,33)(16,43)(18,31,27,39)(20,49)(21,51)(26,37,28,40)(29,47)(30,46)(34,54)(41,59,61,64)(42,58)(44,57)(45,56)(48,53)(50,63,62,55)(52,60);(1,3)(2,9,12,32)(4,20,22,47)(5,16)(6,27)(7,18)(8,17)(10,33)(11,30,38,43)(13,39,14,31)(15,24,42,23)(19,29)(21,50,53,62)(25,46)(26,45)(28,44)(34,59)(35,49)(36,58)(37,57,40,56)(41,52,61,54)(48,55)(51,63)(60,64);(1,4)(2,10)(3,15)(5,21)(6,24)(7,23)(8,22)(9,29)(11,34)(12,36)(13,35)(14,19)(16,41)(17,42)(18,47)(20,27)(25,53)(26,54)(28,52)(30,55)(31,58)(32,49)(33,39)(37,48)(38,60)(40,51)(43,63)(44,50)(45,62)(46,61)(56,59)(57,64);(1,5,6,28,8,25,7,26)(2,11,14,37,12,38,13,40)(3,16,27,44,17,46,18,45)(4,21,24,52,22,53,23,54)(9,30,31,57,32,43,39,56)(10,34,19,48,36,60,35,51)(15,41,20,50,42,61,47,62)(29,55,58,64,49,63,33,59)Polycyclic presentation:
2^9|a^2=h,b^2=efg,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269842);