Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1276 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,6,7,8)(9,10,16,21,14,12,15,19)(11,18,22,23,13,17,20,24);(2,4)(6,8)(9,11,15,20,14,13,16,22)(10,17,19,23,12,18,21,24);(1,3,5,7)(2,6,4,8)(9,12,14,10)(11,17,13,18)(15,19,16,21)(20,24,22,23);(1,4,5,2)(3,6,7,8)(9,13,14,11)(10,18,12,17)(15,20,16,22)(19,23,21,24)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338919, #72338922, #72338942, … |
| maximal normal subgroup of | #72338919, #72338922, #72338942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384366);