Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17242 |
| 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 | |
592 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)(3,13,16,8)(4,18,20,9)(5,22)(6,26)(7,11)(10,24)(12,29)(14,33)(15,45,23,56)(17,32,47,41)(19,50,52,55)(21,35,27,39)(25,40)(28,38)(30,43)(31,46,36,49)(34,53,37,54)(42,61)(44,58)(48,60)(51,64)(57,63)(59,62);(1,3,6,17,7,16,29,47)(2,8,26,41,11,13,12,32)(4,14,21,44,20,43,27,48)(5,23,24,49,25,15,28,46)(9,33,39,58,18,30,35,60)(10,36,40,56,38,31,22,45)(19,51,53,63,52,42,54,62)(34,57,50,61,37,59,55,64);(1,4)(2,9)(3,14,16,43)(5,19)(6,27)(7,20)(8,30,13,33)(10,37)(11,18)(12,39)(15,42,23,51)(17,48,47,44)(21,29)(22,55)(24,54)(25,52)(26,35)(28,53)(31,59,36,57)(32,60,41,58)(34,38)(40,50)(45,61,56,64)(46,63,49,62);(1,5,7,25)(2,10,11,38)(3,15,16,23)(4,19,20,52)(6,28,29,24)(8,31,13,36)(9,34,18,37)(12,40,26,22)(14,42,43,51)(17,49,47,46)(21,54,27,53)(30,57,33,59)(32,45,41,56)(35,50,39,55)(44,63,48,62)(58,61,60,64)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386976);