Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2308 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13794 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,29)(3,14,17,33,18,9,28,32)(4,19,22,10,23,37,21,36)(5,24,26,38,27,11,25,39)(15,41,44,34,45,49,43,30)(16,31,47,58,48,40,46,57)(20,51,52,35,54,50,53,55)(42,61,62,59,63,60,64,56);(1,3)(2,9)(4,15)(5,16)(6,28)(7,17)(8,18)(10,34)(11,31)(12,14)(13,33)(19,49)(20,42)(21,44)(22,43)(23,45)(24,40)(25,47)(26,46)(27,48)(29,32)(30,36)(35,59)(37,41)(38,58)(39,57)(50,61)(51,60)(52,64)(53,62)(54,63)(55,56);(1,4,7,22,8,23,6,21)(2,10,13,19,12,36,29,37)(3,15,17,43,18,45,28,44)(5,20,26,53,27,54,25,52)(9,30,33,41,14,34,32,49)(11,35,38,50,24,55,39,51)(16,42,46,62,48,63,47,64)(31,56,58,60,40,59,57,61);(1,5)(2,11)(3,16)(4,20)(6,25)(7,26)(8,27)(9,31)(10,35)(12,24)(13,38)(14,40)(15,42)(17,46)(18,48)(19,50)(21,52)(22,53)(23,54)(28,47)(29,39)(30,56)(32,57)(33,58)(34,59)(36,55)(37,51)(41,60)(43,62)(44,64)(45,63)(49,61)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,g^b=gi,g^c=gi,g^d=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263122);