Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201958 |
| 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:
(1,2,6,14,7,15,23,34)(3,17,21,39,18,48,50,32)(4,9,8,16,25,12,30,44)(5,26,29,33,28,45,57,43)(10,35,37,58,13,24,31,27)(11,38,41,51,42,19,52,22)(20,36,55,62,49,40,64,59)(46,61,60,54,47,53,63,56);(2,9,15,12)(3,18)(5,27)(10,31,13,37)(11,39,42,32)(14,16,34,44)(17,41,48,52)(19,38)(20,54)(21,50)(22,51)(24,57)(26,43,45,33)(28,58)(29,35)(36,62,40,59)(46,60,47,63)(49,56)(53,55)(61,64);(2,10)(4,23)(5,28)(6,25)(9,31)(11,36)(12,37)(13,15)(14,33)(16,26)(17,46)(19,50)(20,49)(21,38)(24,29)(27,58)(32,59)(34,43)(35,57)(39,62)(40,42)(41,60)(44,45)(47,48)(52,63)(53,55)(54,56)(61,64);(1,3,8,22)(2,11,16,17)(4,19,23,50)(5,20,27,56)(6,21,25,38)(7,18,30,51)(9,32,34,52)(10,36,26,46)(12,39,14,41)(13,40,45,47)(15,42,44,48)(24,53,57,64)(28,49,58,54)(29,55,35,61)(31,59,43,63)(33,60,37,62)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,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,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, 109235);