Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,16,11)(8,9)(10,12)(14,20,17,15)(18,19);(5,7)(6,11)(8,17)(9,15)(10,16)(12,13)(14,18)(19,20);(5,8)(6,9)(7,14,16,17)(10,18)(11,20,13,15)(12,19);(2,4)(5,9)(6,8)(7,15)(10,19)(11,14)(12,18)(13,17)(16,20)Polycyclic presentation:
2^9|a^2=f,c^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72251367, #72251386, #72251390 |
| maximal normal subgroup of | #72251367, #72251386, #72251390 |
| maximal quotient of | #72251502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331805);