Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,22,15)(12,23,25,16)(13,27)(14,17)(18,30)(20,31)(21,32,28,35)(24,40)(26,34,29,36)(33,39)(37,38);(9,11,14,22)(10,15,17,19)(12,24,25,39)(13,28,30,21)(16,33,23,40)(18,35,27,32)(20,29,38,26)(31,36,37,34);(9,12)(10,16)(11,20)(13,29)(14,25)(15,31)(17,23)(18,36)(19,37)(21,39)(22,38)(24,28)(26,30)(27,34)(32,40)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18772);