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 | #62911 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2599 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)(8,13)(14,27)(15,21)(16,24)(17,25)(18,26)(20,23)(22,28)(29,32)(30,31)(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)(34,40)(35,43)(37,46)(38,47)(39,50)(41,53)(42,54)(44,56)(45,57)(48,60)(49,61)(51,62)(52,55)(58,63)(59,64);(1,5,7,18)(2,10,11,26)(3,14,15,30)(4,16,6,17)(8,22,23,32)(9,24,19,25)(12,27,21,31)(13,28,20,29)(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=i,c^2=h,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, 6260042);