Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62909 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,21)(4,9,6,19)(5,10,17,25)(8,20,23,13)(14,26,28,31)(16,24,18,29)(22,30,32,27)(33,34,38,41)(35,39,45,50)(36,40,47,52)(37,42,48,54)(43,55,57,49)(44,51,58,61)(46,53,59,62)(56,63,64,60);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,28)(18,30)(24,31)(25,32)(26,29)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,55)(53,60)(54,61)(59,64)(62,63);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,17,18)(8,21,23,12)(10,24,25,29)(14,27,28,30)(22,31,32,26)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,55)(51,60)(54,62)(58,64)(61,63);(1,5)(2,10)(3,14)(4,16)(6,18)(7,17)(8,22)(9,24)(11,25)(12,26)(13,27)(15,28)(19,29)(20,30)(21,31)(23,32)(34,41)(39,50)(40,52)(42,54)(49,55)(51,61)(53,62)(60,63);(5,17)(10,25)(14,28)(16,18)(22,32)(24,29)(26,31)(27,30)(33,37,38,48)(34,42,41,54)(35,44,45,58)(36,46,47,59)(39,51,50,61)(40,53,52,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260103);