Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1591 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)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,41)(30,38)(32,42)(33,39)(34,40)(35,46)(37,45)(43,48)(44,47);(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,42)(20,29,35,45,33,43,34,44)(25,37,40,41,39,47,46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,44)(31,45)(32,43)(36,46)(38,47)(42,48);(2,8)(7,15)(9,13)(10,16)(20,33)(25,39)(29,43)(34,35)(37,47)(40,46)(41,48)(44,45);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,44,43,45)(37,41,47,48)Polycyclic presentation:
2^9|b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261243);