Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1272 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,11,14,17)(12,19,22,20,23,16,21,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,17)(14,18)(16,22)(19,21)(23,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,d^c=di,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72267801, #72267846, #72280475 |
| maximal normal subgroup of | #72267801, #72267846, #72280475 |
| maximal quotient of | #72267823, #72267862 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18686);