Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15375 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1190 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)(4,19)(5,25)(6,30)(7,12)(8,13)(10,23)(11,27)(14,35)(15,32)(16,40)(17,33)(18,34)(20,49)(21,48)(22,41)(24,37)(26,38)(28,39)(29,42)(31,45)(36,52)(43,63)(44,58)(46,59)(47,57)(50,56)(51,54)(53,61)(55,60)(62,64);(1,3)(2,9)(4,20)(5,26)(6,17)(7,16)(8,18)(10,35)(11,38)(12,40)(13,34)(14,24)(15,27)(19,31)(21,50)(22,45)(23,44)(25,32)(28,46)(29,47)(30,33)(36,60)(37,58)(39,57)(41,49)(42,59)(43,51)(48,56)(52,55)(53,64)(54,63)(61,62);(1,4)(2,10)(3,14)(5,21)(6,24)(7,23)(8,22)(9,31)(11,36)(12,41)(13,37)(15,43)(16,20)(17,45)(18,44)(19,30)(25,54)(26,55)(27,53)(28,52)(29,51)(32,56)(33,58)(34,49)(35,40)(38,62)(39,48)(42,61)(46,64)(47,50)(57,60)(59,63);(1,5,6,29,8,27,7,28)(2,11,12,42,13,25,30,39)(3,15,16,47,18,26,17,46)(4,21,24,51,22,53,23,52)(9,32,33,59,34,38,40,57)(10,36,41,61,37,54,19,48)(14,43,20,50,44,55,45,64)(31,56,58,63,49,62,35,60)Polycyclic presentation:
2^9|d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335037);