Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343554 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,10,7,11,5,9)(3,12,13,4,8,15,14,16)(17,18,22,26,23,27,21,35)(19,28,30,38,31,24,36,39)(20,25,34,40,33,41,32,45)(29,42,43,47,44,37,46,48);(1,3,7,8)(2,4,11,16)(5,13,6,14)(9,15,10,12)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,28)(32,43)(33,44)(34,46)(35,39)(40,47)(41,42)(45,48);(1,4)(2,8)(3,11)(5,12)(6,15)(7,16)(9,13)(10,14)(17,20)(18,25)(19,29)(21,32)(22,34)(23,33)(24,37)(26,40)(27,41)(28,42)(30,43)(31,44)(35,45)(36,46)(38,47)(39,48);(1,5,7,6)(2,9,11,10)(3,13,8,14)(4,15,16,12)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,33,34)(24,38,28,39)(25,40,41,45)(29,43,44,46)(37,47,42,48);(20,33)(25,41)(29,44)(32,34)(37,42)(40,45)(43,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,f^2=i,g^2=hi,b^a=bf,d^a=dh,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251427);