Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #299195 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #33179 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
981 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 981 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10)(11,19)(12,24)(13,28)(14,18)(15,22)(16,26)(17,25)(20,33)(21,34)(23,32)(27,37)(29,36)(30,38)(31,35)(39,40);(9,11,13,22,14,23,25,33)(10,15,17,19,18,20,28,32)(12,21,26,35,27,36,38,40)(16,29,24,31,30,34,37,39);(1,3,4,2,5,6,8,7)(9,10)(11,20,23,15)(12,16)(13,17)(14,18)(19,22,32,33)(21,34,36,29)(24,26)(25,28)(27,30)(31,35,39,40)(37,38);(9,12,25,38,14,27,13,26)(10,16,28,37,18,30,17,24)(11,21,33,40,23,36,22,35)(15,29,32,39,20,34,19,31)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 206394);