Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1490 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 1490 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,7)(4,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,22);(1,2,5,3)(4,7,8,6)(9,10,14,17)(11,19,21,23)(12,18,13,15)(16,22,24,20);(1,3,5,2)(4,6,8,7)(9,11,14,21)(10,16,17,24)(12,20,13,22)(15,19,18,23);(10,17)(15,18)(16,24)(19,23)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214129, #72214656, #72219580 |
| maximal normal subgroup of | #72214129, #72214656, #72219580 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141090);