Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122159 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1064 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 146 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)(11,18)(12,21)(13,17)(14,19)(15,20)(16,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14,17,18)(12,20)(13,19)(15,21,23,16)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,23)(12,20)(13,19)(14,16,18,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,g^b=gh,d^c=di,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260820, #72260837 |
| maximal normal subgroup of | #72260820, #72260837 |
| maximal quotient of | #72261429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29355);