Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366633 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13995 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,14,30)(3,13,18,37,17,8,27,33)(4,19,23,38,22,9,28,35)(5,24,26,39,25,10,29,36)(15,31,43,54,20,40,46,56)(16,32,45,55,44,41,47,57)(21,48,51,59,50,34,52,58)(42,53,61,63,49,60,62,64);(1,3,14,18,7,17,6,27)(2,8,30,33,11,13,12,37)(4,20,28,46,22,15,23,43)(5,16,26,47,25,44,29,45)(9,31,38,54,19,40,35,56)(10,32,36,57,24,41,39,55)(21,49,51,61,50,42,52,62)(34,53,58,64,48,60,59,63);(1,4)(2,9)(3,15)(5,21)(6,28)(7,22)(8,31)(10,34)(11,19)(12,38)(13,40)(14,23)(16,42)(17,20)(18,46)(24,48)(25,50)(26,52)(27,43)(29,51)(30,35)(32,53)(33,56)(36,59)(37,54)(39,58)(41,60)(44,49)(45,62)(47,61)(55,64)(57,63);(1,5)(2,10)(3,16)(4,21)(6,29)(7,25)(8,32)(9,34)(11,24)(12,39)(13,41)(14,26)(15,42)(17,44)(18,47)(19,48)(20,49)(22,50)(23,52)(27,45)(28,51)(30,36)(31,53)(33,57)(35,59)(37,55)(38,58)(40,60)(43,62)(46,61)(54,64)(56,63)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,h^c=hi,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, 273831);