Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129435 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
981 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 981 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 24 points:
(9,10)(11,17)(12,20,21,14)(13,16)(15,19)(18,24,22,23);(1,2,4,6)(3,7,8,5)(10,14,16,20)(12,21)(15,24,17,23)(18,22);(1,3,4,8)(2,5,6,7)(9,11)(10,15,16,17)(12,22)(13,19)(14,23,20,24)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gh,d^c=di,e^c=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319655, #72319712, #72320021, … |
| maximal normal subgroup of | #72319655, #72319712, #72320021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56642);