Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,14,16,10)(8,11)(9,12)(13,17)(15,19,20,18);(1,2)(3,4)(5,7,8,15,9,16,17,20)(6,10,11,18,12,14,13,19);(2,4)(6,11,12,13)(10,18,14,19);(10,14)(18,19)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250313, #72256869, #72256917, … |
| maximal normal subgroup of | #72250313, #72256869, #72256917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357161);