Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#127 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
848 conjugacy classes of subgroups; only the 246 normal subgroups are shown.
This group has 848 conjugacy classes of subgroups — too many to draw. Only its 246 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,14,18,15,19,13,23)(11,20,12,17,22,24,21,16);(9,11,13,21,15,22,14,12)(10,16,18,20,19,17,23,24);(1,2,6,8,4,7,3,5);(9,12,14,22,15,21,13,11)(10,17,18,24,19,16,23,20)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72210332, #72210416, #72210456, … |
| maximal normal subgroup of | #72210332, #72210416, #72210456, … |
| maximal quotient of | #72210308, #72216241 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120433);