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×#538 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13)(3,14,19,45)(4,10,22,38)(5,23,24,39)(6,26,29,42)(7,12,20,36)(9,30,34,15)(11,21,40,54)(16,48,51,31)(17,44,18,35)(25,57,53,37)(27,33,47,32)(28,56,58,41)(43,49,55,64)(46,63,59,60)(50,62,52,61);(1,3,8,19)(2,9,13,34)(4,15,22,30)(5,16,24,51)(6,27,29,47)(7,18,20,17)(10,35,38,44)(11,31,40,48)(12,33,36,32)(14,42,45,26)(21,46,54,59)(23,55,39,43)(25,50,53,52)(28,49,58,64)(37,60,57,63)(41,61,56,62);(1,4,20,29)(2,10,36,42)(3,15,17,47)(5,21,53,58)(6,8,22,7)(9,14,32,44)(11,37,56,23)(12,26,13,38)(16,46,52,64)(18,27,19,30)(24,54,25,28)(31,43,62,63)(33,35,34,45)(39,40,57,41)(48,55,61,60)(49,51,59,50);(1,5)(2,11)(3,16)(4,21)(6,28)(7,25)(8,24)(9,31)(10,37)(12,41)(13,40)(14,43)(15,46)(17,52)(18,50)(19,51)(20,53)(22,54)(23,42)(26,39)(27,49)(29,58)(30,59)(32,62)(33,61)(34,48)(35,60)(36,56)(38,57)(44,63)(45,55)(47,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,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, 153605);