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 | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1026 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)(35,39,45,53)(36,40)(37,48)(38,42)(41,50)(43,51,58,63)(44,59,61,52)(46,56)(47,55)(49,54)(57,64,60,62);(1,3)(2,8,11,23)(4,13)(5,14)(6,16)(7,15)(9,22,17,31)(10,12,21,27)(18,28)(19,29)(20,30)(24,32,25,26)(33,35)(34,39,42,53)(36,43)(37,44)(38,45)(40,51,55,63)(41,52,48,59)(46,57)(47,58)(49,60)(50,61)(54,62,56,64);(1,4,7,18)(2,9,11,17)(3,13,15,28)(5,20,6,19)(8,22,23,31)(10,25,21,24)(12,26,27,32)(14,30,16,29)(33,36,38,47)(34,40,42,55)(35,43,45,58)(37,49,50,46)(39,51,53,63)(41,56,48,54)(44,60,61,57)(52,64,59,62);(4,18)(9,17)(13,28)(19,20)(22,31)(24,25)(26,32)(29,30)(33,37,38,50)(34,41,42,48)(35,44,45,61)(36,46,47,49)(39,52,53,59)(40,54,55,56)(43,57,58,60)(51,62,63,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,g^2=hi,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, 108553);