Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199851 |
| 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 | |
1003 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)(3,12,16,23,14,27,15,8)(4,17,20,9,18,25,19,24)(13,22,29,26,30,32,28,31)(33,34)(35,39,45,54)(36,40)(37,48)(38,42)(41,51)(43,52,60,63)(44,57,49,53)(46,58)(47,56)(50,55)(59,64,61,62);(1,3)(2,8,11,23)(4,13)(5,15)(6,16)(7,14)(9,22,24,32)(10,12,21,27)(17,31,25,26)(18,30)(19,28)(20,29)(33,35)(34,39,42,54)(36,43)(37,49)(38,45)(40,52,56,63)(41,57,48,53)(44,51)(46,61)(47,60)(50,59)(55,64,58,62);(1,4,7,18)(2,9,11,24)(3,13,14,30)(5,19,6,20)(8,22,23,32)(10,25,21,17)(12,26,27,31)(15,28,16,29)(33,36,38,47)(34,40,42,56)(35,43,45,60)(37,50,51,46)(39,52,54,63)(41,58,48,55)(44,61,49,59)(53,64,57,62);(3,14)(4,18)(8,23)(9,24)(12,27)(15,16)(17,25)(19,20)(33,37,38,51)(34,41,42,48)(35,44,45,49)(36,46,47,50)(39,53,54,57)(40,55,56,58)(43,59,60,61)(52,62,63,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107307);