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 | #1879 |
| prime radical | #33716 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,5)(4,7)(6,8)(9,10,15,21,16,22,14,33,17,23,34,40,32,39,13,20)(11,24,25,37,26,38,12,28,27,19,30,18,29,35,31,36);(9,11,16,29,17,27,32,26)(10,18,22,28,23,37,39,36)(12,14,25,34,31,13,30,15)(19,20,38,21,24,33,35,40);(2,5)(3,8)(6,7)(9,12)(10,19)(11,14)(13,27)(15,26)(16,25)(17,31)(18,20)(21,28)(22,38)(23,24)(29,34)(30,32)(33,37)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gh,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44736);