Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#447 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,39)(4,9,18,32)(5,21,23,52)(6,26,24,11)(8,28,30,15)(10,35,37,25)(14,40,41,61)(17,44,27,36)(19,47,45,63)(20,50,46,34)(22,38,53,31)(29,54,55,42)(33,58,57,49)(43,62,51,59)(48,60,64,56);(1,3)(2,8)(4,14)(5,22)(6,27)(7,16)(9,29)(10,36)(11,38)(12,30)(13,21)(15,25)(17,24)(18,41)(19,48)(20,51)(23,53)(26,31)(28,35)(32,55)(33,59)(34,60)(37,44)(39,52)(40,47)(42,49)(43,46)(45,64)(50,56)(54,58)(57,62)(61,63);(5,23)(10,37)(13,39)(15,28)(19,45)(21,52)(22,53)(25,35)(33,57)(36,44)(40,61)(42,54)(47,63)(48,64)(49,58)(59,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,29)(10,33)(11,34)(12,32)(13,40)(15,42)(16,41)(17,43)(21,47)(22,48)(23,45)(24,46)(25,49)(26,50)(27,51)(28,54)(30,55)(31,56)(35,58)(36,59)(37,57)(38,60)(39,61)(44,62)(52,63)(53,64);(4,18)(9,32)(14,41)(19,45)(20,46)(29,55)(33,57)(34,50)(40,61)(42,54)(43,51)(47,63)(48,64)(49,58)(56,60)(59,62)Polycyclic presentation:
2^9|a^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,f^b=fh,g^b=gi,h^b=hi,f^c=fi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530035);