Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,14,10,15,9,11)(7,16,18,20,19,12,17,13);(1,2)(3,4)(6,11)(7,17)(8,9)(13,20)(14,15)(18,19);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,13)(14,20)(15,16);(5,7)(6,13)(8,18)(9,17)(10,19)(11,12)(14,16)(15,20)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265640, #72265655, #72265675, … |
| maximal normal subgroup of | #72265640, #72265655, #72265675, … |
| maximal quotient of | #72238924, #72286423, #72286596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290254);