Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396094 |
| 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 | #23444 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15,16,11)(8,12)(9,13,18,20)(10,14)(17,19);(1,2)(3,4)(5,7)(6,11)(8,9)(10,16)(12,20)(13,19)(14,15)(17,18);(2,4)(5,8)(6,12)(7,9)(10,17)(11,13)(14,19)(15,20)(16,18);(5,9)(6,13,14,20)(7,8)(10,18)(11,19,15,12)(16,17)Polycyclic presentation:
2^9|a^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251367, #72251386, #72251403 |
| maximal normal subgroup of | #72251367, #72251386, #72251403 |
| maximal quotient of | #72251525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303299);