Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356478 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | D4×#95 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×H(3,ℤ/4) ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1863 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)(3,14,19,40)(4,20)(5,11)(6,12)(7,28)(8,13)(9,17,32,18)(10,22)(15,30,43,56)(16,36,45,60)(21,48)(23,34)(24,52,44,31)(25,37)(26,53)(27,38)(29,35)(33,50)(39,46,47,42)(41,55,61,64)(49,57,62,63)(51,58)(54,59);(1,3)(2,9,13,32)(4,15)(5,24,27,44)(6,17)(7,18)(8,19)(10,30,35,56)(11,36)(12,14,28,40)(16,25,45,26)(20,47,34,39)(21,49,51,62)(22,42)(23,43)(29,46)(31,53)(33,57)(37,52)(38,60)(41,50,61,54)(48,55)(58,64)(59,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30)(11,33)(12,34)(13,35)(14,39)(16,41)(17,42)(18,46)(19,43)(20,28)(24,49)(25,50)(26,54)(27,51)(31,55)(32,56)(36,57)(37,58)(38,59)(40,47)(44,62)(45,61)(48,53)(52,64)(60,63);(1,5)(2,11)(3,16,19,45)(4,21)(6,25)(7,26)(8,27)(9,31,32,52)(10,33)(12,37)(13,38)(14,36,40,60)(15,41,43,61)(17,24,18,44)(20,48)(22,50)(23,51)(28,53)(29,54)(30,55,56,64)(34,58)(35,59)(39,57,47,63)(42,49,46,62)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,g^b=gi,f^c=fi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263674);