Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1180 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)(3,14,18,41)(4,10)(5,25)(6,12)(7,30)(8,13)(9,17,33,19)(11,28)(15,44,47,34)(16,32,48,59)(20,31,46,58)(21,51)(22,36)(23,37)(24,38)(26,39,49,43)(27,40)(29,42)(35,53)(45,61,52,57)(50,64,56,60)(54,63)(55,62);(1,3)(2,9)(4,20)(5,26)(6,19)(7,17)(8,18)(10,34)(11,39)(12,41)(13,33)(14,30)(15,23)(16,27)(21,52)(22,47)(24,46)(25,32)(28,48)(29,49)(31,37)(35,61)(36,58)(38,44)(40,43)(42,59)(45,55)(50,53)(51,64)(54,56)(57,63)(60,62);(1,4)(2,10)(3,15,18,47)(5,21)(6,23)(7,22)(8,24)(9,31,33,58)(11,35)(12,37)(13,38)(14,34,41,44)(16,45,48,52)(17,20,19,46)(25,51)(26,56,49,50)(27,54)(28,53)(29,55)(30,36)(32,57,59,61)(39,64,43,60)(40,63)(42,62);(1,5,8,29)(2,11,13,40)(3,16,18,48)(4,21,24,55)(6,28,7,27)(9,32,33,59)(10,35,38,63)(12,42,30,25)(14,43,41,39)(15,45,47,52)(17,49,19,26)(20,50,46,56)(22,54,23,53)(31,57,58,61)(34,60,44,64)(36,51,37,62)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331797);