Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,17,9,18,33,19,34)(4,10,24,37,23,38,22,42)(5,25,28,11,29,40,27,41)(15,35,46,31,47,43,20,50)(16,32,26,44,48,57,49,39)(21,52,56,61,55,62,54,36)(45,64,51,63,53,59,58,60);(1,3)(2,9)(4,20)(5,26)(6,19)(7,17)(8,18)(10,35)(11,39)(12,33)(13,34)(14,30)(15,24)(16,27)(21,53)(22,47)(23,46)(25,57)(28,48)(29,49)(31,37)(32,40)(36,59)(38,43)(41,44)(42,50)(45,55)(51,54)(52,60)(56,58)(61,64)(62,63);(1,4)(2,10,13,38)(3,15)(5,21)(6,24)(7,22)(8,23)(9,31,34,50)(11,36,41,61)(12,42,30,37)(14,43,33,35)(16,45)(17,20)(18,47)(19,46)(25,52,40,62)(26,58)(27,56)(28,54)(29,55)(32,59,57,64)(39,60,44,63)(48,53)(49,51);(1,5,8,29)(2,11,13,41)(3,16,18,48)(4,21,23,55)(6,28,7,27)(9,32,34,57)(10,36,38,61)(12,25,30,40)(14,44,33,39)(15,45,47,53)(17,49,19,26)(20,51,46,58)(22,56,24,54)(31,59,50,64)(35,60,43,63)(37,62,42,52)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^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,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293134);