Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428193 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15379 |
| 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 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,34)(4,21,25,39)(5,27,30,42)(6,13,7,12)(9,19,35,18)(10,26,38,24)(11,31,41,29)(16,32,22,36)(17,47,49,33)(23,52,55,37)(28,40,48,60)(43,45,59,46)(44,64,51,62)(50,56,63,54)(53,61,57,58);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,40)(12,34)(13,15)(14,35)(16,25)(17,29)(21,43)(23,53)(24,45)(26,46)(27,33)(30,48)(31,49)(32,38)(37,62)(39,59)(41,60)(42,47)(44,56)(50,61)(51,54)(52,64)(55,57)(58,63);(1,4,6,24,8,25,7,26)(2,10,12,21,14,38,13,39)(3,16,18,45,20,22,19,46)(5,23,29,54,30,55,31,56)(9,32,15,43,35,36,34,59)(11,37,27,50,41,52,42,63)(17,44,28,57,49,51,48,53)(33,58,40,64,47,61,60,62);(1,5,8,30)(2,11,14,41)(3,17)(4,23,25,55)(6,29,7,31)(9,33)(10,37,38,52)(12,27,13,42)(15,40)(16,44)(18,28)(19,48)(20,49)(21,50,39,63)(22,51)(24,54,26,56)(32,58)(34,60)(35,47)(36,61)(43,64)(45,57)(46,53)(59,62)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=ef,e^2=i,f^2=i,h^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=de,e^b=ei,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335390);