Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
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 | #61038 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
829 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 829 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,8,6)(9,10,13,19,15,21,14,20)(11,22,25,37,27,34,26,40)(12,28,31,18,33,38,32,39)(16,29,35,23,17,30,36,24);(5,8)(6,7)(9,11)(10,16)(12,29)(13,26)(14,25)(15,27)(17,21)(18,37)(19,36)(20,35)(22,38)(23,32)(24,31)(28,34)(30,33)(39,40);(1,3)(2,4)(5,6)(7,8)(9,11)(10,17)(12,30)(13,26)(14,25)(15,27)(16,21)(18,37)(19,35)(20,36)(22,38)(23,31)(24,32)(28,34)(29,33)(39,40)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=g,f^2=ghi,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, 39050);