Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131808 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3273 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 40 points:
(1,2,5,7,4,3,8,6)(9,10)(11,23,26,39)(12,29)(13,18)(14,21)(15,20)(16,22)(17,24,35,32)(19,31)(25,37,40,36)(27,33,28,34)(30,38);(5,8)(6,7)(9,11)(10,17)(12,25)(13,32)(14,28)(15,27)(16,26)(18,33)(19,23)(20,37)(21,36)(22,35)(24,30)(29,39)(31,40)(34,38);(1,3)(2,4)(5,6)(7,8)(10,18)(12,30)(13,31)(14,15)(17,33)(19,20)(21,29)(22,38)(23,37)(24,25)(27,28)(32,40)(34,35)(36,39)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=gi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=eg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39058);