Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,18)(5,22)(6,26)(7,11)(8,16)(9,20)(10,24)(12,30)(14,42)(15,46)(17,33)(19,35)(21,40)(23,56)(25,41)(27,39)(28,36)(29,38)(31,48)(32,47)(34,45)(37,49)(43,57)(44,50)(51,53)(52,58)(54,59)(55,60)(61,63)(62,64);(1,3,6,27,7,16,30,17)(2,8,12,39,11,13,26,33)(4,14,20,48,21,44,28,45)(5,23,24,47,25,15,29,49)(9,34,18,50,36,31,40,42)(10,37,22,46,38,32,41,56)(19,52,53,62,54,43,55,63)(35,57,60,64,59,58,51,61);(1,4)(2,9)(3,14,16,44)(5,19)(6,28)(7,21)(8,31,13,34)(10,35)(11,36)(12,40)(15,43,23,52)(17,48,27,45)(18,26)(20,30)(22,51)(24,55)(25,54)(29,53)(32,57,37,58)(33,42,39,50)(38,59)(41,60)(46,64,56,61)(47,63,49,62);(1,5,7,25)(2,10,11,38)(3,15,16,23)(4,19,21,54)(6,29,30,24)(8,32,13,37)(9,35,36,59)(12,41,26,22)(14,43,44,52)(17,49,27,47)(18,51,40,60)(20,55,28,53)(31,57,34,58)(33,56,39,46)(42,61,50,64)(45,63,48,62);(2,11)(8,13)(9,36)(10,38)(12,26)(18,40)(22,41)(31,34)(32,37)(33,39)(35,59)(42,50)(46,56)(51,60)(57,58)(61,64)Polycyclic presentation:
2^9|b^2=fg,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411873);