Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13979 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,19,33,18,32,17,9)(4,20,22,10,24,36,23,37)(5,25,27,38,28,55,26,11)(15,30,42,49,43,34,48,39)(16,44,47,57,46,31,45,40)(21,51,52,50,54,35,53,59)(41,60,61,56,62,64,63,58);(1,3)(2,9,13,33)(4,15,24,43)(5,16)(6,17)(7,19)(8,18)(10,34)(11,31,38,44)(12,14,29,32)(20,49)(21,41,54,62)(22,42,23,48)(25,40,55,57)(26,45)(27,47)(28,46)(30,37)(35,58)(36,39)(50,64)(51,56)(52,61,53,63)(59,60);(1,4)(2,10,13,37)(3,15,18,43)(5,21)(6,22)(7,23)(8,24)(9,30)(11,35,38,51)(12,20,29,36)(14,39)(16,41,46,62)(17,42,19,48)(25,50,55,59)(26,52)(27,53)(28,54)(31,56)(32,49)(33,34)(40,60)(44,58)(45,61,47,63)(57,64);(1,5,6,26,8,28,7,27)(2,11,12,25,13,38,29,55)(3,16,17,45,18,46,19,47)(4,21,22,52,24,54,23,53)(9,31,14,40,33,44,32,57)(10,35,20,50,37,51,36,59)(15,41,42,61,43,62,48,63)(30,56,39,60,34,58,49,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272975);