Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500850 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3058 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #28097 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
632 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)(6,12)(7,11)(8,13)(10,22)(14,30)(15,43)(16,34)(17,33)(18,35)(20,50)(21,39)(23,40)(24,37)(26,32)(27,55)(28,57)(29,41)(31,45)(36,47)(38,51)(42,58)(44,60)(46,59)(48,62)(49,52)(53,63)(54,64)(56,61);(1,3,14,42,7,16,29,48,8,18,27,56,5,26,6,17)(2,9,30,58,11,34,41,62,13,35,55,61,25,32,12,33)(4,20,51,44,24,47,23,46,21,52,28,15,22,54,53,45)(10,36,63,59,19,49,38,43,37,64,40,31,39,50,57,60);(1,4)(2,10)(3,15)(5,24)(6,28)(7,22)(8,21)(9,31)(11,39)(12,40)(13,37)(14,23)(16,46)(17,47)(18,44)(19,25)(20,42)(26,45)(27,53)(29,51)(30,38)(32,60)(33,49)(34,43)(35,59)(36,58)(41,63)(48,54)(50,62)(52,56)(55,57)(61,64);(1,2)(3,9)(4,10)(5,25)(6,12)(7,11)(8,13)(14,30)(15,31)(16,34)(17,33)(18,35)(19,24)(20,36)(21,37)(22,39)(23,38)(26,32)(27,55)(28,40)(29,41)(42,58)(43,46)(44,59)(45,60)(47,49)(48,62)(50,54)(51,63)(52,64)(53,57)(56,61)Polycyclic presentation:
2^9|b^2=egh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72217290, #72320748, #72342989, … |
| maximal normal subgroup of | #72217290, #72320748, #72342989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413405);