Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130074 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #45991 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
677 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 677 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,5,6,7)(9,10,13,18,14,16,11,19)(12,15,20,23,21,17,22,24);(1,2)(3,7)(4,6)(5,8)(10,15)(11,13)(12,21)(16,17)(18,23)(19,24);(9,11,14,13)(10,16)(12,20,21,22)(18,19)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal quotient of | #72268606, #72268607 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37336);