Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#459 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2013 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)(4,19)(5,11)(6,27)(7,29)(8,14)(9,17)(10,15)(12,28)(16,39)(18,41)(20,47)(21,40)(22,48)(23,31)(24,37)(25,53)(26,54)(30,34)(32,46)(33,42)(35,43)(36,44)(38,52)(45,60)(49,56)(50,61)(51,64)(55,58)(57,62)(59,63);(1,3,6,17,8,18,28,7)(2,9,12,13,14,29,27,41)(4,15,21,43,23,44,22,30)(5,16,25,46,24,45,52,26)(10,31,34,40,36,19,35,48)(11,32,38,39,37,54,53,60)(20,42,50,63,49,62,51,55)(33,56,58,61,57,47,59,64);(1,4)(2,10)(3,15)(5,20)(6,21)(7,30)(8,23)(9,31)(11,33)(12,34)(13,40)(14,36)(16,42)(17,43)(18,44)(19,29)(22,28)(24,49)(25,50)(26,55)(27,35)(32,56)(37,57)(38,58)(39,61)(41,48)(45,62)(46,63)(47,54)(51,52)(53,59)(60,64);(1,5)(2,11)(3,16)(4,20)(6,25)(7,26)(8,24)(9,32)(10,33)(12,38)(13,39)(14,37)(15,42)(17,46)(18,45)(19,47)(21,50)(22,51)(23,49)(27,53)(28,52)(29,54)(30,55)(31,56)(34,58)(35,59)(36,57)(40,61)(41,60)(43,63)(44,62)(48,64);(5,24)(11,37)(16,45)(20,49)(25,52)(26,46)(32,54)(33,57)(38,53)(39,60)(42,62)(47,56)(50,51)(55,63)(58,59)(61,64)Polycyclic presentation:
2^9|b^2=f,f^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530149);