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 | #2815 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,16)(3,13,21,55)(4,23,26,51)(5,27,30,46)(6,31,34,52)(8,15,17,14)(9,36,41,20)(10,24,45,18)(11,28,42,62)(12,49,53,35)(19,44,29,38)(22,50,56,54)(25,48,33,37)(32,40,63,39)(43,58,47,61)(57,64,59,60);(1,3,17,56)(2,9,14,39)(4,24,30,29)(5,19,26,18)(6,32,49,36)(7,21,8,22)(10,42,44,37)(11,38,48,45)(12,50,52,55)(13,53,54,31)(15,40,16,41)(20,34,63,35)(23,60,46,43)(25,58,62,57)(27,47,51,64)(28,59,33,61);(1,4,8,5)(2,10,15,38)(3,18,22,29)(6,33,35,62)(7,26,17,30)(9,37,40,11)(12,47,31,60)(13,23,50,27)(14,44,16,45)(19,21,24,56)(20,59,32,58)(25,49,28,34)(36,57,63,61)(39,42,41,48)(43,52,64,53)(46,55,51,54);(1,5,8,4)(2,11,15,37)(3,19,22,24)(6,25,35,28)(7,30,17,26)(9,38,40,10)(12,51,31,46)(13,43,50,64)(14,48,16,42)(18,21,29,56)(20,59,32,58)(23,52,27,53)(33,49,62,34)(36,57,63,61)(39,45,41,44)(47,54,60,55)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,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, 153586);