Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| 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 | C22×D4 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 120 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)(9,10,14,20,16,22,15,21)(11,23,25,37,12,27,26,38)(13,29,32,19,34,39,33,40)(17,30,35,24,18,31,36,28);(5,8)(6,7)(9,11,16,12)(10,17,22,18)(13,30,34,31)(14,26,15,25)(19,37,40,38)(20,36,21,35)(23,29,27,39)(24,32,28,33);(1,3)(2,4)(5,6)(7,8)(9,12,16,11)(10,18,22,17)(13,31,34,30)(14,25,15,26)(19,38,40,37)(20,35,21,36)(23,39,27,29)(24,33,28,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,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=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309598, #72309623, #72309625, … |
| maximal normal subgroup of | #72309598, #72309623, #72309625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39065);