Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,23)(13,16)(14,17)(15,21)(19,24)(20,22);(1,3,4,8)(2,5,6,7)(9,11,13,19,14,21,23,22)(10,15,12,24,17,18,16,20);(2,6)(5,7)(11,19,21,22)(15,20,18,24);(3,8)(5,7)(9,12,13,10,14,16,23,17)(11,20,19,18,21,24,22,15)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=ef,d^2=efg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227423 |
| maximal normal subgroup of | #72227423 |
| maximal quotient of | #72268632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361120);