Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16562 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
917 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)(3,15)(4,21)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30,42,29)(16,44)(17,48,52,33)(19,35)(20,34)(22,36)(23,53,55,37)(24,39)(26,38)(28,40,54,61)(32,47)(43,50,56,51)(45,60,57,64)(46,59)(49,58,63,62);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,16,26,46,25,47)(5,17,30,51,31,52,29,50)(10,36,38,32,39,59,21,44)(11,33,41,43,42,48,27,56)(23,49,54,45,55,63,28,57)(37,60,61,58,53,64,40,62);(1,4,6,25,8,26,7,24)(2,10,12,21,14,39,13,38)(3,16,18,22,20,47,19,46)(5,28,30,55,31,54,29,23)(9,32,15,36,35,44,34,59)(11,40,41,53,42,61,27,37)(17,49,51,57,52,63,50,45)(33,60,43,62,48,64,56,58);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,41)(13,27)(14,42)(15,43)(16,45)(18,51)(19,50)(20,52)(21,40)(22,49)(24,54)(25,28)(26,55)(32,58)(34,56)(35,48)(36,60)(38,61)(39,53)(44,62)(46,63)(47,57)(59,64)Polycyclic presentation:
2^9|a^2=eg,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374439);