Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10,12,17,14,18,13,23)(11,19,21,24,22,15,20,16);(10,15,18,19)(11,20)(12,13)(16,17,24,23)(21,22);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215297, #72216446, #72216453, … |
| maximal normal subgroup of | #72215297, #72216446, #72216453, … |
| maximal quotient of | #72265305, #72280500, #72286552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53706);