Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(10,16)(11,12)(13,14)(17,18)(19,23)(21,22);(1,2,4,6)(3,7,8,5)(9,10,15,19)(11,20,22,24)(12,18,21,17)(13,23,14,16);(1,3,4,8)(2,5,6,7)(9,11,13,21,15,22,14,12)(10,17,16,20,19,18,23,24);(9,12,14,22,15,21,13,11)(10,18,16,24,19,17,23,20)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224750, #72224756, #72224757 |
| maximal normal subgroup of | #72224750, #72224756, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318235);