Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201978 |
| 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 | C2×#121 | C23 |
| 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) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,7,15,24,34)(3,17,22,39,18,47,49,32)(4,9,8,16,26,12,19,41)(5,27,30,33,29,45,57,44)(10,35,37,51,13,25,31,28)(11,38,42,50,43,20,52,23)(21,36,55,60,48,40,64,59)(46,53,63,56,58,62,61,54);(2,9,15,12)(3,18)(5,28)(10,31,13,37)(11,39,43,32)(14,16,34,41)(17,42,47,52)(20,38)(21,54)(22,49)(23,50)(25,57)(27,44,45,33)(29,51)(30,35)(36,60,40,59)(46,63,58,61)(48,56)(53,55)(62,64);(2,10)(3,18)(4,24)(5,29)(6,26)(9,31)(11,40)(12,37)(13,15)(14,33)(16,27)(17,46)(20,22)(23,50)(25,30)(28,51)(32,60)(34,44)(35,57)(36,43)(38,49)(39,59)(41,45)(42,63)(47,58)(52,61)(53,64)(55,62);(1,3,19,50)(2,11,41,47)(4,20,6,22)(5,21,51,54)(7,18,8,23)(9,32,14,42)(10,36,45,46)(12,39,34,52)(13,40,27,58)(15,43,16,17)(24,49,26,38)(25,53,30,55)(28,56,29,48)(31,59,33,61)(35,62,57,64)(37,60,44,63)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109255);