Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #534 | C2×C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,14,19)(11,20,25,38)(12,26,21,17)(13,28)(15,22,34,23)(16,30)(18,31)(24,35)(27,36)(29,32)(33,39)(37,40);(2,6,8,7)(9,11,21,22,14,25,12,23)(10,15,26,38,19,34,17,20)(13,29,30,40,31,24,27,39)(16,35,28,37,36,32,18,33);(1,3,5,4)(2,6,8,7)(9,12,14,21)(10,16,19,36)(11,22,25,23)(13,27,31,30)(15,32,34,35)(17,18,26,28)(20,37,38,33)(24,39,29,40)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46833);