Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478349 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17142 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,38)(3,13,16,39,17,8,24,30)(4,9,21,34,20,33,25,40)(5,10,22,37,23,36,26,41)(14,42,18,50,45,28,48,31)(15,43,46,59,47,29,49,58)(19,35,54,63,53,32,55,62)(44,56,52,61,64,57,51,60);(1,3,7,17)(2,8,11,13)(4,18,20,48)(5,15,23,47)(6,24,27,16)(9,31,33,50)(10,29,36,43)(12,39,38,30)(14,25,45,21)(19,52,53,51)(22,49,26,46)(28,40,42,34)(32,61,35,60)(37,59,41,58)(44,55,64,54)(56,63,57,62);(1,4)(2,9,11,33)(3,14)(5,19)(6,25)(7,20)(8,28,13,42)(10,35,36,32)(12,40,38,34)(15,44)(16,48)(17,45)(18,24)(21,27)(22,55)(23,53)(26,54)(29,57,43,56)(30,50,39,31)(37,62,41,63)(46,51)(47,64)(49,52)(58,61,59,60);(1,5)(2,10,11,36)(3,15)(4,19)(6,26)(7,23)(8,29,13,43)(9,32,33,35)(12,41,38,37)(14,44)(16,49)(17,47)(18,51)(20,53)(21,55)(22,27)(24,46)(25,54)(28,56,42,57)(30,59,39,58)(31,60,50,61)(34,63,40,62)(45,64)(48,52)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385546);