Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,9,15)(3,17,21,37)(4,22,20,48)(5,25,30,12)(6,14,31,50)(7,34,8,16)(10,35,39,18)(11,41,45,24)(13,29,47,27)(19,52,54,46)(23,55,26,40)(28,36,62,42)(32,63,33,51)(38,60,44,53)(43,58,57,56)(49,61,64,59);(1,3,9,21)(2,10,15,39)(4,8,20,7)(5,26,30,23)(6,24,31,41)(11,42,45,36)(12,37,25,17)(13,48,47,22)(14,43,50,57)(16,40,34,55)(18,29,35,27)(19,33,54,32)(28,60,62,53)(38,51,44,63)(46,64,52,49)(56,61,58,59);(1,4,8,21)(2,11,16,46)(3,9,20,7)(5,26,27,18)(6,24,33,54)(10,36,40,49)(12,43,47,44)(13,38,25,57)(14,22,51,37)(15,45,34,52)(17,50,48,63)(19,31,41,32)(23,29,35,30)(28,58,59,53)(39,42,55,64)(56,61,60,62);(2,12)(5,27)(6,31)(10,37)(11,43)(13,34)(14,36)(15,25)(16,47)(17,39)(18,26)(19,54)(22,40)(23,35)(24,41)(28,61)(29,30)(32,33)(38,52)(42,50)(44,46)(45,57)(48,55)(49,51)(53,56)(58,60)(59,62)(63,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153581);