Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14030 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,30)(3,14,18,33,19,41,17,9)(4,20,24,10,25,36,23,37)(5,26,29,38,28,55,27,11)(15,31,44,39,21,34,43,57)(16,45,47,61,48,32,46,40)(22,35,52,54,53,59,51,49)(42,60,63,58,50,64,62,56);(1,3)(2,9,13,33)(4,21,25,15)(5,16)(6,17)(7,19)(8,18)(10,34)(11,32,38,45)(12,14,30,41)(20,39)(22,50,53,42)(23,44,24,43)(26,40,55,61)(27,46)(28,48)(29,47)(31,37)(35,58)(36,57)(49,60)(51,63,52,62)(54,64)(56,59);(1,4)(2,10)(3,15,19,21)(5,22)(6,23)(7,25)(8,24)(9,31,33,34)(11,35)(12,36)(13,37)(14,39,41,57)(16,42,48,50)(17,43,18,44)(20,30)(26,54)(27,51)(28,53)(29,52)(32,56,45,58)(38,59)(40,60,61,64)(46,62,47,63)(49,55);(1,5,6,27,7,28,8,29)(2,11,12,26,13,38,30,55)(3,16,17,46,19,48,18,47)(4,22,23,51,25,53,24,52)(9,32,14,40,33,45,41,61)(10,35,36,54,37,59,20,49)(15,42,43,62,21,50,44,63)(31,56,39,60,34,58,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=efg,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 276751);