Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15312 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 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)(4,18)(5,10)(6,26)(7,12)(8,16)(9,22)(11,30)(14,44)(15,43)(17,39)(19,42)(20,34)(21,35)(23,57)(24,41)(25,38)(27,33)(28,40)(29,37)(31,46)(32,49)(36,47)(45,59)(48,50)(51,55)(52,62)(53,64)(54,61)(56,60)(58,63);(1,3,7,17)(2,8,12,33)(4,19,22,14)(5,23,25,49)(6,27,30,16)(9,31,18,50)(10,36,38,43)(11,39,26,13)(15,29,47,24)(20,53,55,63)(21,46,28,48)(32,41,57,37)(34,59,51,62)(35,42,40,44)(45,54,52,56)(58,61,64,60);(1,4,6,28,7,22,30,21)(2,9,11,40,12,18,26,35)(3,14,16,48,17,19,27,46)(5,20,29,54,25,55,24,56)(8,31,13,42,33,50,39,44)(10,34,41,60,38,51,37,61)(15,45,49,63,47,52,23,53)(32,58,43,59,57,64,36,62);(1,5)(2,10)(3,15)(4,20)(6,29)(7,25)(8,32)(9,34)(11,41)(12,38)(13,43)(14,45)(16,49)(17,47)(18,51)(19,52)(21,56)(22,55)(23,27)(24,30)(26,37)(28,54)(31,58)(33,57)(35,61)(36,39)(40,60)(42,59)(44,62)(46,53)(48,63)(50,64)Polycyclic presentation:
2^9|b^2=i,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330366);