Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200122 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60954 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,12,18)(3,13,15,44)(4,17)(5,23)(9,32,36,64)(10,29,30,49)(11,38,41,58)(14,45,19,43)(16,25,47,61)(21,54)(22,50)(24,52,59,31)(26,27)(28,40,46,42)(33,63,37,62)(34,57,51,35)(39,56)(48,55,53,60);(1,2,4,18,20,30,50,29)(3,9,19,51,47,36,48,34)(5,24,21,38,23,41,56,31)(6,10,17,49,7,12,22,8)(11,39,52,27,59,54,58,26)(13,42,45,64,61,40,60,32)(14,46,16,37,53,28,15,33)(25,57,55,63,44,35,43,62);(1,3)(2,9)(4,19)(5,25)(6,15)(7,16)(8,28)(10,33)(11,40)(12,37)(13,27)(14,17)(18,51)(20,47)(21,55)(22,53)(23,44)(24,57)(26,61)(29,34)(30,36)(31,62)(32,52)(35,41)(38,63)(39,60)(42,59)(43,56)(45,54)(46,49)(48,50)(58,64);(2,10)(8,29)(9,33)(11,24)(12,30)(18,49)(28,34)(31,58)(32,63)(35,42)(36,37)(38,52)(40,57)(41,59)(46,51)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107318);