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 | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,11)(4,15)(5,18)(6,10)(7,13)(8,16)(9,21)(12,23)(14,25)(17,26)(19,22)(20,24)(27,28)(29,31)(30,32)(33,34,38,42)(35,39,45,52)(36,40,47,54)(37,48,49,41)(43,55,57,50)(44,58,59,51)(46,53,60,61)(56,63,64,62);(1,3)(2,7)(4,12)(5,19)(6,14)(8,23)(9,25)(10,22)(11,18)(13,21)(15,27)(16,30)(17,29)(20,28)(24,31)(26,32)(33,35,38,45)(34,39,42,52)(36,43,47,57)(37,44,49,59)(40,50,54,55)(41,51,48,58)(46,56,60,64)(53,62,61,63);(7,22)(11,25)(23,31)(27,32)(33,36,38,47)(34,40,42,54)(35,43,45,57)(37,46,49,60)(39,50,52,55)(41,53,48,61)(44,56,59,64)(51,62,58,63);(1,4,6,17)(2,8,10,24)(3,12,14,29)(5,20,21,16)(7,23,22,31)(9,26,18,15)(11,27,25,32)(13,30,19,28)(33,37)(34,41)(35,44)(36,46)(38,49)(39,51)(40,53)(42,48)(43,56)(45,59)(47,60)(50,62)(52,58)(54,61)(55,63)(57,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^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, 153607);