Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456520 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16283 |
| 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 | |
1105 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)(5,27,30,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,44)(17,43,50,61)(19,35)(20,34)(22,32)(23,39,53,56)(24,38)(26,37)(28,36,54,62)(29,40,31,41)(33,49,55,51)(42,47)(45,58,52,63)(46,59)(48,64,57,60);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,47,26,16,25,46)(5,17,31,51,30,50,29,49)(10,32,37,59,38,44,21,42)(11,33,41,43,27,55,40,61)(23,52,54,48,53,45,28,57)(36,58,56,64,62,63,39,60);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,39)(12,21)(13,37)(14,38)(15,42)(17,48)(18,47)(19,46)(20,22)(23,29)(27,56)(30,54)(31,53)(33,60)(34,59)(35,44)(36,40)(41,62)(43,63)(45,49)(50,57)(51,52)(55,64)(58,61);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,36)(12,41)(13,40)(14,27)(15,43)(16,45)(18,51)(19,49)(20,50)(21,39)(22,52)(24,54)(25,28)(26,53)(32,58)(34,61)(35,55)(37,56)(38,62)(42,60)(44,63)(46,57)(47,48)(59,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363717);