Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131796 |
| 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 | #61089 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
863 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 863 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)(9,10,14,21,16,23,15,22)(11,24,27,33,29,40,28,36)(12,20,13,19,32,17,30,34)(18,35,37,26,39,25,38,31);(5,8)(6,7)(10,17)(12,30)(13,32)(14,15)(18,33)(19,21)(20,23)(22,34)(24,37)(25,26)(27,28)(31,35)(36,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,11)(10,18)(12,31)(13,25)(14,28)(15,27)(16,29)(17,33)(19,40)(20,36)(21,38)(22,37)(23,39)(24,34)(26,32)(30,35)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=gi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,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=eg,f^c=fi,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, 39053);