Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622850 |
| 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 ◁ #3573 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3841 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)(3,16)(4,11)(5,25)(6,14)(7,13)(8,15)(9,18)(10,19)(12,28)(17,44)(20,42)(21,37)(22,50)(23,40)(24,39)(26,43)(27,31)(29,41)(30,36)(32,46)(33,55)(34,48)(35,47)(38,53)(45,61)(49,60)(51,62)(52,56)(54,59)(57,63)(58,64);(1,3)(2,9)(4,17)(5,26)(6,19)(7,30)(8,20)(10,31)(11,32)(12,36)(13,33)(14,43)(15,34)(16,25)(18,28)(21,45)(22,51)(23,47)(24,54)(27,48)(29,55)(35,56)(37,57)(38,59)(39,58)(40,62)(41,42)(44,50)(46,53)(49,64)(52,63)(60,61);(1,3)(2,10)(4,17)(5,18)(6,30)(7,19)(8,20)(9,31)(11,35)(12,34)(13,42)(14,16)(15,36)(21,45)(22,46)(23,54)(24,47)(25,43)(26,28)(27,55)(29,48)(32,56)(33,41)(37,59)(38,57)(39,61)(40,44)(49,63)(50,62)(51,53)(52,64)(58,60);(1,4)(2,11)(3,17)(5,22)(6,23)(7,24)(8,21)(9,32)(10,35)(12,38)(13,39)(14,40)(15,37)(16,44)(18,46)(19,47)(20,45)(25,50)(26,51)(27,52)(28,53)(29,49)(30,54)(31,56)(33,58)(34,57)(36,59)(41,60)(42,61)(43,62)(48,63)(55,64);(4,21)(11,37)(17,45)(22,49)(23,24)(32,57)(35,59)(38,56)(39,40)(44,61)(46,63)(47,54)(50,60)(51,64)(52,53)(58,62)Polycyclic presentation:
2^9|f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,f^b=fh,h^b=hi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530046);