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 | #62934 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2598 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)(5,10,18,26)(8,20,23,13)(14,27,30,31)(16,24,17,25)(22,29,32,28)(33,34,38,42)(35,39,45,52)(36,40,47,54)(37,48,49,41)(43,55,57,50)(44,58,59,51)(46,60,61,53)(56,62,64,63);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,28)(17,29)(18,30)(24,31)(25,27)(26,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,50)(41,51)(42,52)(46,56)(47,57)(48,58)(49,59)(53,62)(54,55)(60,63)(61,64);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,18,17)(8,21,23,12)(10,24,26,25)(14,28,30,29)(22,31,32,27)(33,36,38,47)(34,40,42,54)(35,43,45,57)(37,46,49,61)(39,50,52,55)(41,53,48,60)(44,56,59,64)(51,62,58,63);(1,5)(2,10)(3,14)(4,16)(6,17)(7,18)(8,22)(9,24)(11,26)(12,27)(13,28)(15,30)(19,25)(20,29)(21,31)(23,32)(33,37,38,49)(34,41,42,48)(35,44,45,59)(36,46,47,61)(39,51,52,58)(40,53,54,60)(43,56,57,64)(50,62,55,63);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,17,18,16)(8,21,23,12)(10,25,26,24)(14,29,30,28)(22,27,32,31)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=i,e^2=h,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, 6260131);