Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458577 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| 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 | |
956 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)(3,9,17,31)(4,10)(5,23)(6,14)(7,13)(8,12)(11,25)(15,42,44,29)(16,46,48,59)(18,40,27,32)(19,33,45,57)(20,52)(21,41)(22,35)(24,38)(26,39)(28,36)(30,50,58,49)(34,53)(37,55)(43,62,47,60)(51,63,64,56)(54,61);(1,3)(2,9,12,31)(4,19)(5,16)(6,27)(7,18)(8,17)(10,33,35,57)(11,30,38,58)(13,40,14,32)(15,21)(20,51)(22,45)(23,46,39,59)(24,50)(25,49)(26,48)(28,44)(29,41,42,36)(34,60,37,62)(43,55)(47,53)(52,63,61,56)(54,64);(1,4)(2,10)(3,15,17,44)(5,20)(6,21)(7,28)(8,22)(9,29,31,42)(11,37)(12,35)(13,36)(14,41)(16,47,48,43)(18,19,27,45)(23,52)(24,53)(25,55)(26,54)(30,56,58,63)(32,33,40,57)(34,38)(39,61)(46,62,59,60)(49,51,50,64);(1,5,8,26)(2,11,12,38)(3,16,17,48)(4,20,22,54)(6,25,7,24)(9,30,31,58)(10,34,35,37)(13,23,14,39)(15,43,44,47)(18,50,27,49)(19,51,45,64)(21,55,28,53)(29,56,42,63)(32,59,40,46)(33,60,57,62)(36,61,41,52)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365774);