Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10645 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1440 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7,12)(3,13,14,16)(4,8,15,9)(5,11,6,10)(17,18)(19,27,29,23)(20,24,32,41)(21,34)(22,25)(26,37)(28,38)(30,39,35,43)(31,48)(33,44,36,42)(40,47)(45,46);(1,3)(2,8)(4,6)(5,15)(7,14)(9,12)(10,16)(11,13)(17,19)(18,23)(20,31)(21,35)(22,29)(24,40)(25,27)(26,43)(28,33)(30,37)(32,47)(34,39)(36,46)(38,42)(41,48)(44,45);(1,4,5,14,7,15,6,3)(2,9,10,16,12,8,11,13)(17,20)(18,24,25,41)(19,28)(21,36)(22,32)(23,38,27,45)(26,44,34,42)(29,46)(30,47)(31,35)(33,37)(39,48,43,40)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26128);