Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #371873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14082 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1020 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)(3,8,15,29)(4,17)(5,10,23,33)(6,25)(7,11)(9,20)(12,22)(13,35,37,26)(14,38)(16,42,39,28)(18,30,47,43)(19,31,50,44)(21,32)(24,52,51,34)(27,40)(36,57)(41,54)(45,48)(46,55)(49,62,64,56)(53,58)(59,60)(61,63);(1,3)(2,8,11,29)(4,18,20,47)(5,14)(6,16)(7,15)(9,30)(10,27,33,38)(12,28,25,42)(13,21,37,48)(17,43)(19,46,50,63)(22,39)(23,40)(24,41)(26,45)(31,55)(32,35)(34,54,52,59)(36,49,58,64)(44,61)(51,60)(53,62)(56,57);(1,4)(2,9,11,17)(3,13,15,37)(5,19)(6,21)(7,20)(8,26)(10,31,33,44)(12,32,25,45)(14,36,40,58)(16,18,39,47)(22,48)(23,50)(24,49)(27,53)(28,30)(29,35)(34,56,52,62)(38,57)(41,46,60,63)(42,43)(51,64)(54,55)(59,61);(1,5,22,51,7,23,6,24)(2,10,25,52,11,33,12,34)(3,14,39,60,15,40,16,41)(4,19,48,64,20,50,21,49)(8,27,42,59,29,38,28,54)(9,31,45,62,17,44,32,56)(13,36,47,63,37,58,18,46)(26,53,43,61,35,57,30,55)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=gi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279070);