Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363956 |
| 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 | |
2447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,9,19,32,17,31,18,30)(4,20,23,10,24,36,22,35)(5,11,26,37,25,38,27,39)(15,40,44,28,42,54,43,33)(16,45,46,56,47,29,48,55)(21,49,51,58,50,59,52,34)(41,61,62,53,63,60,64,57);(1,3)(2,9,12,31)(4,15)(5,16)(6,19)(7,18)(8,17)(10,33,35,28)(11,29,38,45)(13,32,14,30)(20,40,36,54)(21,41)(22,44)(23,43)(24,42)(25,47)(26,48)(27,46)(34,57,58,53)(37,56,39,55)(49,60,59,61)(50,63)(51,64)(52,62);(1,4,8,24)(2,10,12,35)(3,15,17,42)(5,21,25,50)(6,23,7,22)(9,28,31,33)(11,34,38,58)(13,36,14,20)(16,41,47,63)(18,44,19,43)(26,52,27,51)(29,53,45,57)(30,54,32,40)(37,59,39,49)(46,64,48,62)(55,61,56,60);(1,5)(2,11,12,38)(3,16)(4,21)(6,26)(7,27)(8,25)(9,29,31,45)(10,34,35,58)(13,37,14,39)(15,41)(17,47)(18,46)(19,48)(20,49,36,59)(22,51)(23,52)(24,50)(28,53,33,57)(30,55,32,56)(40,60,54,61)(42,63)(43,62)(44,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271155);