Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,3)(9,10)(11,15,16,13)(12,14);(1,2,4,3)(5,6)(7,8);(1,3,4,2)(6,8)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72261268, #72261411, #72262130, … |
| maximal normal subgroup of | #72261268, #72261411, #72262130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303104);