Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347398 |
| 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 | #62829 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,26)(4,9,18,24)(5,11,6,10)(8,20,22,15)(14,27,29,31)(17,25,19,23)(21,28,32,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,55)(44,51)(45,50)(46,53)(47,52)(48,54)(49,57)(56,63)(58,61)(59,62)(60,64);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,32)(25,27)(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)(2,9)(3,14)(5,19)(6,17)(7,18)(8,21)(10,25)(11,23)(12,24)(13,27)(15,30)(16,29)(20,28)(22,32)(26,31)(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);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,17,18,19)(8,13,22,26)(9,23,24,25)(14,28,29,30)(21,31,32,27)(34,41)(39,50)(40,52)(42,54)(49,55)(51,61)(53,62)(60,63);(3,16)(8,22)(13,26)(14,29)(15,20)(21,32)(27,31)(28,30)(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=h,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254616);