Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141475 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
460 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(2,3)(4,5)(7,8)(9,10)(11,18)(12,16,19,15)(13,17)(14,20,23,24)(21,22);(1,2,4,3,6,8,5,7)(9,11,19,24,13,21,12,20)(10,14,15,18,17,23,16,22);(1,3,4,8,6,7,5,2)(9,12)(10,15)(11,20)(13,19)(14,22)(16,17)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=egi,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48712);