Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107114 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2536 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 2536 conjugacy classes of subgroups — too many to draw. Only its 206 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)(11,17)(12,15)(13,16)(14,19)(18,21)(20,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14,16,17)(12,18)(13,19)(15,21,22,23)(20,24);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72218180, #72218206, #72225610, … |
| maximal normal subgroup of | #72218180, #72218206, #72225610, … |
| maximal quotient of | #72218854, #72236816, #72236978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14310);