Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C24×C4 | C23 |
| commutator | C23 | C2×C4×C8 |
| socle | C25 | C22×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
3874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(17,18,19,21,20,22,23,24);(2,4)(9,10)(11,12)(13,14,15,16);(6,8)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72314119, #72314156, #72314179, … |
| maximal normal subgroup of | #72314119, #72314156, #72314179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30703);