Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #15412 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
925 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 925 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,8,5,3,4,7)(9,10)(11,17,19,13)(12,15)(14,16)(18,23,24,20)(21,22);(1,3,5,2)(4,7,6,8)(9,11,13,16)(10,14,19,17)(12,18,20,22)(15,21,24,23);(4,6)(7,8)(9,12,14,21)(10,15,16,22)(11,18,19,24)(13,20,17,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72234501, #72234508, #72234509, … |
| maximal normal subgroup of | #72234501, #72234508, #72234509, … |
| maximal quotient of | #72283699, #72284550, #72338749 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46701);