Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
764 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 764 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(2,6)(7,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,24)(22,23);(1,2,5,6)(3,7,4,8)(10,15,18,17)(16,19,24,23);(1,3,5,4)(2,7,6,8)(9,11,12,21,14,22,13,20)(10,16,15,23,18,24,17,19);(1,4,5,3)(2,7,6,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=efg,c^2=egh,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi74 conjugacy classes — view the full character table.
| maximal quotient of | #72235145, #72235146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361139);