Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#127 |
| prime radical | #19616 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
798 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 218 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)(3,7)(6,8)(9,10,13,17,14,18,22,16)(11,19,12,15,20,24,21,23);(9,11,13,12,14,20,22,21)(10,15,16,19,18,23,17,24);(1,2,3,5,4,6,7,8)(11,20)(12,21)(15,23)(19,24);(2,6)(5,8)(9,12,13,20,14,21,22,11)(10,15,17,24,18,23,16,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=dhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224816, #72224924, #72224926 |
| maximal normal subgroup of | #72224816, #72224924, #72224926 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120520);