Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814819 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,28)(20,26)(21,25)(22,24)(23,31)(27,36)(29,38)(30,44)(32,39)(33,48)(34,41)(35,40)(37,47)(42,46)(43,45);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,23)(20,33)(21,31)(22,29)(24,39)(25,28)(26,42)(27,38)(30,45)(32,36)(34,46)(35,47)(37,43)(40,44)(41,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,24,25,27)(19,29,31,32)(20,34)(23,28)(26,43,41,40)(30,46,47,33)(35,45)(38,39);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,25)(23,28)(24,27)(26,41)(37,44)(38,39)(40,43)(42,48);(3,13)(7,10)(11,12)(15,16)(17,20,22,35,21,34,36,45)(18,26,24,40,25,41,27,43)(19,30,32,46,31,47,29,33)(23,37,38,48,28,44,39,42)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722938);