Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| 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 | |
1374 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)(3,14,17,9,18,35,19,34)(4,20,23,38,24,10,25,39)(5,26,28,11,30,41,29,42)(15,43,21,32,47,36,46,61)(16,48,50,44,49,33,27,40)(22,53,57,51,56,58,55,37)(45,64,52,63,54,60,59,62);(1,3)(2,9)(4,21)(5,27)(6,19)(7,17)(8,18)(10,36)(11,40)(12,35)(13,34)(14,31)(15,25)(16,28)(20,43)(22,54)(23,47)(24,46)(26,33)(29,49)(30,50)(32,38)(37,60)(39,61)(41,48)(42,44)(45,56)(51,64)(52,55)(53,62)(57,59)(58,63);(1,4)(2,10,13,20)(3,15)(5,22)(6,25)(7,23)(8,24)(9,32,34,61)(11,37,42,51)(12,39,31,38)(14,43,35,36)(16,45)(17,21)(18,47)(19,46)(26,58,41,53)(27,59)(28,57)(29,55)(30,56)(33,60,48,64)(40,62,44,63)(49,54)(50,52);(1,5)(2,11)(3,16)(4,22)(6,29)(7,28)(8,30)(9,33)(10,37)(12,41)(13,42)(14,44)(15,45)(17,50)(18,49)(19,27)(20,51)(21,52)(23,57)(24,56)(25,55)(26,31)(32,60)(34,48)(35,40)(36,62)(38,58)(39,53)(43,63)(46,59)(47,54)(61,64)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293119);