Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,17,9,19,34,18,35)(4,20,23,39,25,38,24,10)(5,26,28,41,30,11,29,42)(15,45,21,43,48,32,47,36)(16,40,49,60,50,44,27,33)(22,51,55,57,56,62,54,37)(46,64,52,63,53,59,58,61);(1,3)(2,9)(4,21)(5,27)(6,18)(7,17)(8,19)(10,36)(11,40)(12,34)(13,35)(14,31)(15,24)(16,28)(20,45)(22,53)(23,48)(25,47)(26,44)(29,50)(30,49)(32,38)(33,41)(37,59)(39,43)(42,60)(46,56)(51,61)(52,54)(55,58)(57,64)(62,63);(1,4)(2,10,13,39)(3,15)(5,22)(6,24)(7,23)(8,25)(9,32,35,45)(11,37,26,57)(12,20,31,38)(14,43,34,36)(16,46)(17,21)(18,47)(19,48)(27,58)(28,55)(29,54)(30,56)(33,59,60,64)(40,61,44,63)(41,62,42,51)(49,52)(50,53);(1,5)(2,11)(3,16)(4,22)(6,29)(7,28)(8,30)(9,33)(10,37)(12,42)(13,26)(14,44)(15,46)(17,49)(18,27)(19,50)(20,51)(21,52)(23,55)(24,54)(25,56)(31,41)(32,59)(34,40)(35,60)(36,61)(38,62)(39,57)(43,63)(45,64)(47,58)(48,53)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293127);