Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13956 |
| 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 | |
2455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,13,6,12)(3,15,18,31,19,38,27,9)(4,10,21,34,22,35,26,37)(5,23,24,36,25,39,28,11)(16,40,44,32,45,56,43,29)(17,46,49,41,48,30,47,57)(20,50,53,33,52,54,51,59)(42,55,61,64,62,58,63,60);(1,3)(2,9,13,31)(4,16)(5,17)(6,18)(7,27)(8,19)(10,32,35,29)(11,30,36,46)(12,15,14,38)(20,42)(21,43)(22,45)(23,41,39,57)(24,47)(25,48)(26,44)(28,49)(33,58,59,55)(34,40,37,56)(50,60,54,64)(51,61)(52,62)(53,63);(1,4,8,22)(2,10,13,35)(3,16,19,45)(5,20,25,52)(6,26,7,21)(9,29,31,32)(11,33,36,59)(12,37,14,34)(15,40,38,56)(17,42,48,62)(18,44,27,43)(23,54,39,50)(24,53,28,51)(30,55,46,58)(41,60,57,64)(47,63,49,61);(1,5)(2,11,13,36)(3,17)(4,20)(6,24)(7,28)(8,25)(9,30,31,46)(10,33,35,59)(12,23,14,39)(15,41,38,57)(16,42)(18,47)(19,48)(21,51)(22,52)(26,53)(27,49)(29,55,32,58)(34,50,37,54)(40,60,56,64)(43,61)(44,63)(45,62)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,h^b=hi,e^c=ei,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, 271104);