Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470811 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1586 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,27)(20,31)(21,35)(22,25)(23,30)(24,34)(26,36)(28,43)(29,42)(32,41)(33,40)(37,48)(38,46)(39,45)(44,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,21,30,22,29,36,46)(18,23,26,27,25,38,35,42)(20,32,33,48,34,28,44,45)(24,39,40,43,31,37,47,41);(3,9)(6,13)(10,11)(14,15)(17,20,21,33,22,34,36,44)(18,24,26,40,25,31,35,47)(19,28,30,45,29,32,46,48)(23,37,27,41,38,39,42,43);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(19,29)(23,38)(27,42)(28,32)(30,46)(37,39)(41,43)(45,48);(2,8)(6,13)(7,16)(14,15)(18,25)(19,29)(24,31)(26,35)(28,32)(30,46)(40,47)(45,48)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6389392);