Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1023 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)(12,20,21,15)(13,16)(14,18,22,24)(19,23);(1,2,4,6)(3,7,8,5)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24);(1,3,4,8)(2,5,6,7)(12,21)(15,20)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,g^b=gh,d^c=di,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72281733, #72281836, #72281865, … |
| maximal normal subgroup of | #72281733, #72281836, #72281865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56630);