Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6)(7,8)(9,10)(11,15,16,13)(12,14);(1,2,3,4);(9,11)(10,13)(12,16)(14,15);(6,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=h,b^2=e,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^c=gi,h^c=hi,f^d=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72281532, #72281558, C4×#32926, … |
| maximal normal subgroup of | #72281532, #72281558, C4×#32926, … |
| maximal quotient of | #72312894, #72313299, #72313916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299364);