Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1228 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)(5,6,7,12,8,13,15,20,9,10,14,17,16,11,18,19);(6,10)(7,14)(11,13)(15,18);(2,4);(6,11,10,13)(7,14)(12,20,17,19)(15,18)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^b=gi,g^d=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214871, #72214893, #72219586, … |
| maximal normal subgroup of | #72214871, #72214893, #72219586, … |
| maximal quotient of | #72219601, #72239474 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401141);