Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2308 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13784 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1413 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)(3,14,18,9)(4,20)(5,26,28,11)(6,29)(7,13)(8,12)(10,24)(15,41,21,48)(16,32)(17,33,19,40)(22,49,52,59)(23,50)(25,35)(27,36,30,37)(31,43,38,44)(34,53,55,54)(39,46)(42,61)(45,58)(47,57)(51,64)(56,62)(60,63);(1,3,7,19,8,18,6,17)(2,9,13,33,12,14,29,40)(4,21,25,43,23,15,24,44)(5,16,27,45,28,46,30,47)(10,31,20,48,35,38,50,41)(11,32,37,58,26,39,36,57)(22,51,53,63,52,42,54,62)(34,56,49,64,55,60,59,61);(1,4,6,24,8,23,7,25)(2,10,13,20,12,35,29,50)(3,15,19,44,18,21,17,43)(5,22,30,54,28,52,27,53)(9,31,40,41,14,38,33,48)(11,34,37,49,26,55,36,59)(16,42,45,62,46,51,47,63)(32,56,57,61,39,60,58,64);(1,5)(2,11)(3,16)(4,22)(6,30)(7,27)(8,28)(9,32)(10,34)(12,26)(13,37)(14,39)(15,42)(17,47)(18,46)(19,45)(20,49)(21,51)(23,52)(24,54)(25,53)(29,36)(31,56)(33,58)(35,55)(38,60)(40,57)(41,61)(43,63)(44,62)(48,64)(50,59)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=ce,g^b=gi,e^c=ei,f^d=fi,g^d=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262686);