Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15245 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
954 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)(3,14)(4,10)(5,24)(6,28)(7,12)(8,13)(9,17)(11,26)(15,39)(16,44)(18,32)(19,41)(20,30)(21,34)(22,36)(23,37)(25,38)(27,54)(29,35)(31,47)(33,43)(40,46)(42,61)(45,52)(48,57)(49,56)(50,60)(51,53)(55,59)(58,62)(63,64);(1,3,7,18,8,19,6,17)(2,9,28,41,13,32,12,14)(4,20,22,43,23,48,29,15)(5,16,27,47,26,46,25,45)(10,33,35,30,37,39,36,57)(11,31,54,44,24,52,38,40)(21,49,50,62,51,63,55,42)(34,58,59,56,53,61,60,64);(1,4)(2,10)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30)(11,34)(12,35)(13,37)(14,39)(16,42)(17,20)(18,48)(19,43)(24,53)(25,50)(26,51)(27,55)(28,36)(31,56)(32,57)(33,41)(38,59)(40,61)(44,58)(45,49)(46,62)(47,63)(52,64)(54,60);(1,5,6,25,8,26,7,27)(2,11,28,54,13,24,12,38)(3,16,17,45,19,46,18,47)(4,21,22,50,23,51,29,55)(9,31,41,44,32,52,14,40)(10,34,36,60,37,53,35,59)(15,42,20,49,43,62,48,63)(30,56,33,58,57,64,39,61)Polycyclic presentation:
2^9|b^2=eh,d^2=gh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324464);