Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,9)(8,11,16,19)(10,12)(13,15,20,18)(14,17);(1,3,4,2)(5,7)(6,9)(8,13,16,20)(10,14)(11,18,19,15)(12,17);(5,8)(6,11,12,19)(7,13,14,20)(9,15)(10,16)(17,18);(5,9)(6,7)(8,15)(10,17)(11,13)(12,14)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225628, #72225638, #72225639, … |
| maximal normal subgroup of | #72225628, #72225638, #72225639, … |
| maximal quotient of | #72251578, #72265805, #72269857 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275435);