Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15453 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1308 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)(3,9,16,31)(4,10)(5,24)(6,13)(7,28)(8,12)(11,27)(14,40,42,29)(15,44,45,57)(17,32,18,48)(19,49,43,33)(20,51)(21,36)(22,35)(23,56)(25,39)(26,38)(30,46,37,47)(34,54)(41,59,63,64)(50,62,52,58)(53,60)(55,61);(1,3)(2,9,12,31)(4,19,21,43)(5,15,25,45)(6,17)(7,18)(8,16)(10,33)(11,37)(13,32,28,48)(14,22,42,23)(20,52)(24,57)(26,46,27,47)(29,56)(30,38)(34,59,60,64)(35,40)(36,49)(39,44)(41,54)(50,55)(51,58,61,62)(53,63);(1,4)(2,10)(3,14,16,42)(5,20)(6,22)(7,23)(8,21)(9,29,31,40)(11,34)(12,36)(13,35)(15,41,45,63)(17,19,18,43)(24,51)(25,55)(26,53)(27,54)(28,56)(30,58,37,62)(32,33,48,49)(38,60)(39,61)(44,64,57,59)(46,52,47,50);(1,5,8,25)(2,11,12,38)(3,15)(4,20,21,55)(6,27,7,26)(9,30)(10,34,36,60)(13,39,28,24)(14,41)(16,45)(17,47)(18,46)(19,50)(22,54,23,53)(29,58)(31,37)(32,57)(33,59)(35,61,56,51)(40,62)(42,63)(43,52)(44,48)(49,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339438);