Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427835 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,8,16,33)(4,17,19,39)(5,21,23,10)(6,25,29,11)(9,27,35,20)(13,41,43,48)(14,31,22,36)(15,38,26,32)(18,49,51,59)(24,37,28,40)(30,46,56,44)(34,50,53,52)(42,62,54,61)(45,58,47,57)(55,64,60,63);(1,3)(2,8)(4,13)(5,22)(6,26)(7,16)(9,30)(10,36)(11,38)(12,33)(14,23)(15,29)(17,48)(18,42)(19,43)(20,46)(21,31)(24,45)(25,32)(27,44)(28,47)(34,55)(35,56)(37,57)(39,41)(40,58)(49,62)(50,64)(51,54)(52,63)(53,60)(59,61);(1,4)(2,9)(3,13)(5,18)(6,27)(7,19)(8,30)(10,34)(11,39)(12,35)(14,42)(15,46)(16,43)(17,25)(20,29)(21,53)(22,54)(23,51)(24,52)(26,44)(28,50)(31,55)(32,48)(33,56)(36,60)(37,49)(38,41)(40,59)(45,64)(47,63)(57,61)(58,62);(1,5,6,28,7,23,29,24)(2,10,11,40,12,21,25,37)(3,14,15,47,16,22,26,45)(4,18,27,50,19,51,20,52)(8,31,32,58,33,36,38,57)(9,34,39,59,35,53,17,49)(13,42,46,63,43,54,44,64)(30,55,48,62,56,60,41,61)Polycyclic presentation:
2^9|a^2=i,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 335032);