Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139709 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 732 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5,7,8)(4,6)(9,10,11,17,14,19,12,15)(13,18,21,16,22,23,20,24);(1,3)(2,5,6,8)(4,7)(10,15,19,17)(13,22)(16,23,24,18)(20,21);(9,11,14,12)(10,16,19,24)(13,21,22,20)(15,23,17,18)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dgi,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72268115, #72268116 |
| maximal normal subgroup of | #72268115, #72268116 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46907);