Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,14,8,11)(7,15,17,12,18,20,16,19);(2,4)(6,11,14,13)(7,16,18,17)(12,19)(15,20);(5,7)(6,12)(8,16)(9,17)(10,18)(11,15)(13,20)(14,19);(2,4)(6,11,14,13)(12,15,19,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72338478, #72338481, #72338490 |
| maximal normal subgroup of | #72338478, #72338481, #72338490 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280107);