Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,33,19,34)(4,21,24,10,26,37,25,38)(5,27,28,11,30,39,29,40)(16,42,44,58,45,31,22,35)(17,32,47,56,48,57,46,41)(23,36,52,54,53,60,51,49)(43,59,62,61,63,64,50,55);(1,3,7,18,8,20,6,19)(2,9,12,33,14,34,13,15)(4,22,24,16,26,44,25,45)(5,17,28,47,30,48,29,46)(10,35,37,42,38,58,21,31)(11,32,39,56,40,57,27,41)(23,50,52,43,53,62,51,63)(36,59,54,61,60,64,49,55);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,21)(13,37)(14,38)(15,35)(17,43)(18,22)(19,44)(20,45)(27,54)(28,51)(29,52)(30,53)(32,55)(33,58)(34,42)(39,49)(40,60)(41,59)(46,62)(47,50)(48,63)(56,64)(57,61);(1,5,8,30)(2,11,14,40)(3,17,20,48)(4,23,26,53)(6,29,7,28)(9,32,34,57)(10,36,38,60)(12,39,13,27)(15,41,33,56)(16,43,45,63)(18,47,19,46)(21,49,37,54)(22,50,44,62)(24,52,25,51)(31,55,42,61)(35,59,58,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72287080, #72287261, #72287280, … |
| maximal normal subgroup of | #72287080, #72287261, #72287280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271799);