Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,15,27,14,8)(4,17,20,25,18,9,19,24)(13,26,30,22,28,32,29,31)(33,34)(35,44,47,55)(36,48,50,41)(37,51)(38,42)(39,43)(40,46,56,52)(45,59)(49,57,53,58)(54,61)(60,63)(62,64);(1,3,7,15)(2,8)(4,13,18,28)(5,14,6,16)(9,22)(10,12)(11,23)(17,31)(19,29,20,30)(21,27)(24,32)(25,26)(33,35,38,52,39,47,37,46)(34,40,42,55,43,56,51,44)(36,45,53,61,50,62,49,60)(41,54,57,64,48,63,58,59);(1,4)(2,9)(3,13)(5,20)(6,19)(7,18)(8,22)(10,25)(11,17)(12,26)(14,30)(15,28)(16,29)(21,24)(23,31)(27,32)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,48)(44,59)(46,60)(47,62)(51,58)(52,61)(55,64)(56,63);(4,18)(9,17)(13,28)(19,20)(22,31)(24,25)(26,32)(29,30)(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,49,50,53)(40,55,56,44)(41,57,48,58)(45,60,62,61)(54,64,63,59)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108377);