Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(10,16)(11,18)(13,14)(17,23)(19,24)(21,22);(5,6)(7,8)(9,10)(11,13)(12,17)(14,16)(15,18)(19,21)(20,24)(22,23);(2,4)(6,8)(9,11,13,10,15,16,14,18)(12,19,21,17,20,23,22,24);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24);(12,20)(17,24)(19,23)(21,22)Polycyclic presentation:
2^9|c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^b=hi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72316526, #72316532, #72316551, … |
| maximal normal subgroup of | #72316526, #72316532, #72316551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339164);