Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13993 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,18,34,19,33,17,9)(4,20,24,10,25,37,23,38)(5,26,28,39,29,56,27,11)(15,42,21,35,45,31,44,40)(16,46,48,58,49,32,47,41)(22,36,53,55,54,60,52,50)(43,59,51,57,64,61,63,62);(1,3)(2,9,13,34)(4,21,25,44)(5,16)(6,17)(7,18)(8,19)(10,35)(11,32,39,46)(12,14,30,33)(15,24,45,23)(20,42)(22,51,54,63)(26,41,56,58)(27,47)(28,48)(29,49)(31,37)(36,59)(38,40)(43,53,64,52)(50,62)(55,57)(60,61);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,38)(14,40)(16,43)(17,44)(18,21)(19,45)(20,30)(26,55)(27,52)(28,53)(29,54)(32,57)(33,35)(34,42)(39,60)(41,61)(46,62)(47,63)(48,51)(49,64)(50,56)(58,59);(1,5,6,27,8,29,7,28)(2,11,12,26,13,39,30,56)(3,16,17,47,19,49,18,48)(4,22,23,52,25,54,24,53)(9,32,14,41,34,46,33,58)(10,36,37,55,38,60,20,50)(15,43,44,63,45,64,21,51)(31,57,40,61,42,62,35,59)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273652);