Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1889 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,25,35)(3,13,18,38,16,37,45,60)(4,9,21,33,19,31,24,50)(5,10,14,27,23,34,40,54)(8,26,30,44,29,52,57,17)(15,36,42,58,43,59,47,62)(20,32,39,53,22,48,49,63)(28,51,56,46,55,61,64,41);(1,3,14,26,7,16,40,52)(2,8,27,38,12,29,54,60)(4,15,39,51,19,43,49,61)(5,17,6,18,23,44,25,45)(9,28,53,58,31,55,63,62)(10,13,11,30,34,37,35,57)(20,41,21,42,22,46,24,47)(32,36,33,56,48,59,50,64);(1,4)(2,9)(3,15)(5,22)(6,24)(7,19)(8,28)(10,32)(11,33)(12,31)(13,36)(14,39)(16,43)(17,46)(18,47)(20,23)(21,25)(26,51)(27,53)(29,55)(30,56)(34,48)(35,50)(37,59)(38,58)(40,49)(41,44)(42,45)(52,61)(54,63)(57,64)(60,62);(3,16)(4,19)(8,29)(9,31)(13,37)(17,44)(18,45)(20,22)(21,24)(26,52)(30,57)(32,48)(33,50)(38,60)(39,49)(53,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303182, #72303192, #72303219, … |
| maximal normal subgroup of | #72303182, #72303192, #72303219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401785);