Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232147 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1524 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1524 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,12,9,13,17,11)(7,14,15,19,16,10,18,20);(1,2)(3,4)(5,7)(6,10,13,14)(8,15)(9,16)(11,19,12,20)(17,18);(2,4)(6,11)(8,17)(10,19)(12,13)(14,20)(15,18);(7,15,16,18)(10,20,14,19)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=e,f^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279685 |
| maximal normal subgroup of | #72279685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299060);