Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1654 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)(3,13)(4,9,20,33)(5,21)(6,24)(7,12)(8,16)(10,22)(11,27)(14,37,41,57)(15,29)(17,35)(18,45,47,58)(19,48,51,32)(23,36)(25,30)(26,34)(28,49,53,40)(31,50,56,46)(38,42)(39,52,61,64)(43,55)(44,54)(59,62,63,60);(1,3)(2,8)(4,14,20,41)(5,15)(6,25)(7,17)(9,28,33,53)(10,29)(11,35)(12,30)(13,24)(16,27)(18,39,47,61)(19,49,51,40)(21,38)(22,44)(23,43)(26,42)(31,52,56,64)(32,57,48,37)(34,55)(36,54)(45,59,58,63)(46,62,50,60);(1,4)(2,9,12,33)(3,14,17,41)(5,18)(6,19)(7,20)(8,28)(10,31,34,56)(11,32,24,48)(13,37)(15,39,43,61)(16,40,25,49)(21,45,36,58)(22,46)(23,47)(26,50)(27,51)(29,52)(30,53)(35,57)(38,59)(42,60,44,62)(54,63)(55,64);(1,5,7,23)(2,10,12,34)(3,15,17,43)(4,18,20,47)(6,26,27,22)(8,29,30,55)(9,31,33,56)(11,36,24,21)(13,38,35,54)(14,39,41,61)(16,44,25,42)(19,50,51,46)(28,52,53,64)(32,58,48,45)(37,59,57,63)(40,62,49,60)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276081);