Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #61435 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
596 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(10,17,19,18)(11,21,20,22)(12,23)(13,24)(14,15);(1,2)(3,7)(4,6)(5,8)(9,10,16,19)(11,15,20,14)(12,22,13,21)(17,23,18,24);(9,11)(10,15)(12,18)(13,17)(14,19)(16,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=g,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 | #72287269, #72287358, #72287363, … |
| maximal normal subgroup of | #72287269, #72287358, #72287363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39010);