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 | #61086 | 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 | |
798 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 798 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,8,6)(10,17,20,32)(11,23,25,39)(12,28)(13,29)(14,15)(18,34)(19,22,33,21)(24,27,30,26)(31,40)(36,37);(5,8)(6,7)(9,10)(11,18)(12,19)(13,17)(14,22)(15,21)(16,20)(23,38)(24,40)(25,37)(26,34)(27,36)(28,32)(29,33)(30,31)(35,39);(1,3)(2,4)(5,6)(7,8)(9,11,16,25)(10,18,20,37)(12,23,29,39)(13,30,28,24)(14,27,15,26)(17,31,32,40)(19,38,33,35)(21,34,22,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,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=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, 39052);