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 | #3460 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2573 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,27)(15,22)(16,24)(17,26)(18,25)(20,23)(21,29)(28,31)(30,32)(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)(17,30)(18,29)(24,27)(25,32)(26,31)(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);(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);(1,4,7,17)(2,9,11,26)(3,13,15,30)(5,18,6,16)(8,21,23,31)(10,25,19,24)(12,27,22,32)(14,29,20,28)(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);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,27,31,32)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,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, 6260052);