Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6379185 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,18,21,9)(5,10)(6,25)(7,12)(8,17)(11,28)(14,41,43,39)(15,40)(16,37)(19,48,52,32)(20,33,55,49)(22,56)(23,38)(24,36)(26,31)(27,35)(29,44,47,53)(30,46)(34,45)(42,58,62,61)(50,57,63,64)(51,59,54,60);(1,3)(2,8)(4,14)(5,22)(6,26)(7,16)(9,29)(10,34)(11,37)(12,31)(13,25)(15,23)(17,28)(18,47)(19,50)(20,53)(21,43)(24,46)(27,45)(30,35)(32,58)(33,41)(36,40)(38,56)(39,49)(42,51)(44,55)(48,61)(52,63)(54,62)(57,59)(60,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,29)(10,32)(11,33)(12,18)(13,39)(15,42)(16,43)(17,44)(22,50)(23,51)(24,52)(25,49)(26,53)(27,54)(28,55)(30,57)(31,47)(34,58)(35,59)(36,48)(37,41)(38,60)(40,61)(45,62)(46,63)(56,64);(1,5,7,24)(2,10,12,36)(3,15,16,45)(4,19,21,52)(6,27,28,23)(8,30,31,56)(9,32,18,48)(11,38,25,35)(13,40,37,34)(14,42,43,62)(17,46,26,22)(20,54,55,51)(29,57,47,64)(33,60,49,59)(39,61,41,58)(44,63,53,50);(3,16)(8,31)(13,37)(14,43)(15,45)(17,26)(22,46)(29,47)(30,56)(34,40)(39,41)(42,62)(44,53)(50,63)(57,64)(58,61)Polycyclic presentation:
2^9|a^2=h,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,e^b=ei,f^b=fi,h^c=hi,f^d=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286382);