Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,6,4,5,7,8)(13,16);(2,5)(3,7)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,15)(14,16);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fi,g^b=gh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72231778, #72233910, #72238111, … |
| maximal normal subgroup of | #72231778, #72233910, #72238111, … |
| maximal quotient of | #72231765, #72233908, #72233909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95157);