Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| 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 | |
1648 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,41)(30,42)(31,38)(33,39)(34,40)(35,46)(37,44)(43,47)(45,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,38,36,42)(20,29,35,44,33,45,34,43)(25,37,40,41,39,47,46,48);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,45)(31,43)(32,44)(36,46)(38,47)(42,48);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(19,30)(20,33)(24,38)(25,39)(28,42)(31,32)(34,35)(40,46);(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,43,45,44)(37,41,47,48)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,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, 6261882);