Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374124 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2374 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)(3,13)(4,9,21,34)(5,22)(6,25)(7,12)(8,16)(10,23)(11,28)(14,40,18,38)(15,30)(17,36)(19,48,50,59)(20,51,53,33)(24,37)(26,31)(27,35)(29,42,55,47)(32,52,58,49)(39,43)(41,62,46,54)(44,57)(45,56)(60,63,61,64);(1,3)(2,8)(4,18)(5,15)(6,26)(7,17)(9,29)(10,30)(11,36)(12,31)(13,25)(14,21)(16,28)(19,46)(20,42)(22,39)(23,45)(24,44)(27,43)(32,54)(33,40)(34,55)(35,57)(37,56)(38,51)(41,50)(47,53)(48,60)(49,63)(52,64)(58,62)(59,61);(1,4)(2,9,12,34)(3,14,17,18)(5,19)(6,20)(7,21)(8,29)(10,32,35,58)(11,33,25,51)(13,38)(15,41,44,46)(16,42,26,47)(22,48,37,59)(23,49)(24,50)(27,52)(28,53)(30,54)(31,55)(36,40)(39,60)(43,63,45,64)(56,61)(57,62);(1,5,7,24)(2,10,12,35)(3,15,17,44)(4,19,21,50)(6,27,28,23)(8,30,31,57)(9,32,34,58)(11,37,25,22)(13,39,36,56)(14,41,18,46)(16,45,26,43)(20,52,53,49)(29,54,55,62)(33,59,51,48)(38,60,40,61)(42,64,47,63)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72306448, #72306450, #72306453, … |
| maximal normal subgroup of | #72306448, #72306450, #72306453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281321);