Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,16,25)(4,17,18,26)(5,11,6,10)(8,20,23,15)(9,21,24,19)(14,22,28,32)(27,29,31,30)(33,34,38,42)(35,43,45,58)(36,40,48,57)(37,49,52,41)(39,50,54,46)(44,60,61,64)(47,59,51,56)(53,62,55,63);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,22)(10,25)(11,13)(12,23)(17,31)(18,28)(19,30)(21,29)(24,32)(26,27)(33,35)(34,39)(36,44)(37,50)(38,45)(40,55)(41,58)(42,54)(43,49)(46,52)(47,63)(48,61)(51,62)(53,57)(56,64)(59,60);(1,4,7,18)(2,9,12,24)(3,14,16,28)(5,19,6,21)(8,22,23,32)(10,17,11,26)(13,27,25,31)(15,29,20,30)(33,36,38,48)(34,40,42,57)(35,44,45,61)(37,51,52,47)(39,53,54,55)(41,59,49,56)(43,60,58,64)(46,63,50,62);(4,18)(9,24)(14,28)(17,26)(19,21)(22,32)(27,31)(29,30)(35,45)(39,54)(43,58)(44,61)(46,50)(53,55)(60,64)(62,63)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=ei,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142610);