Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14719 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1569 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)(4,18)(5,10)(6,24)(7,12)(8,16)(9,21)(11,25)(14,34)(15,35)(17,36)(19,30)(20,31)(22,48)(23,33)(26,38)(27,40)(28,42)(29,52)(32,49)(37,58)(39,54)(41,57)(43,46)(44,47)(45,62)(50,64)(51,61)(53,59)(55,63)(56,60);(1,3,6,16,7,17,25,42)(2,8,11,13,12,28,24,36)(4,14,20,38,21,39,47,52)(5,15,22,40,23,41,49,61)(9,29,31,54,18,26,44,34)(10,27,32,35,33,51,48,57)(19,37,45,59,46,60,63,64)(30,53,55,58,43,50,62,56);(1,4,6,20,7,21,25,47)(2,9,11,31,12,18,24,44)(3,14,16,38,17,39,42,52)(5,19,22,45,23,46,49,63)(8,26,13,34,28,29,36,54)(10,30,32,55,33,43,48,62)(15,37,40,59,41,60,61,64)(27,50,35,56,51,53,57,58);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,27)(9,30)(11,32)(12,33)(13,35)(14,37)(16,40)(17,41)(18,43)(20,45)(21,46)(24,48)(25,49)(26,50)(28,51)(29,53)(31,55)(34,56)(36,57)(38,59)(39,60)(42,61)(44,62)(47,63)(52,64)(54,58)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301952);