Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480662 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,16)(12,17)(13,21,22,15)(14,18)(19,24,20,23);(2,6)(3,4)(10,15)(11,12)(16,24)(17,23)(18,21)(19,20);(1,3,5,4)(2,7,6,8)(9,11,14,12)(10,16,18,17)(13,19,22,20)(15,23,21,24);(2,6)(7,8)(9,12,14,11)(10,17,18,16)(13,20,22,19)(15,24,21,23)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,f^b=fg,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318066, #72318844, #72318895, … |
| maximal normal subgroup of | #72318066, #72318844, #72318895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387915);