Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16249 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2138 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)(4,20)(5,11)(6,30)(7,12)(8,13)(9,17)(10,23)(15,31)(16,48)(18,33)(19,41)(21,34)(22,37)(24,36)(25,42)(26,35)(27,39)(28,38)(29,40)(32,50)(43,46)(44,52)(45,56)(47,57)(49,60)(51,59)(53,55)(54,61)(58,63)(62,64);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,32)(12,41)(13,33)(14,30)(15,24)(20,31)(22,49)(23,47)(25,46)(26,45)(27,51)(28,50)(29,52)(35,60)(36,43)(37,62)(38,44)(39,48)(40,59)(42,57)(53,56)(54,64)(55,63)(58,61);(1,4)(2,10)(3,15)(5,26)(6,24)(7,23)(8,25)(9,31)(11,37)(12,42)(13,36)(14,43)(16,49)(17,21)(18,46)(19,47)(20,30)(22,27)(28,55)(29,54)(32,58)(33,57)(34,41)(35,38)(39,61)(40,53)(44,62)(45,50)(48,56)(51,64)(52,63)(59,60);(1,5)(2,11)(3,16)(4,22)(6,28)(7,27)(8,29)(9,32)(10,35)(12,39)(13,40)(14,44)(15,45)(17,51)(18,50)(19,52)(20,53)(21,49)(23,54)(24,26)(25,55)(30,38)(31,56)(33,59)(34,60)(36,61)(37,42)(41,48)(43,58)(46,63)(47,64)(57,62)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361305);