Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | #61446 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #2965 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,20,16,22,15,21)(11,23,26,38,27,35,12,28)(13,31,32,19,34,39,33,40)(17,29,18,24,37,30,36,25);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,14)(13,29)(15,26)(16,27)(18,21)(19,38)(20,36)(22,37)(23,39)(24,33)(25,32)(28,40)(30,34)(31,35);(9,12,16,26)(10,18,22,36)(11,14,27,15)(13,24,34,25)(17,21,37,20)(19,23,40,35)(28,31,38,39)(29,33,30,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,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=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38991);