Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1886 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)(7,10,14,17)(8,11)(9,12)(13,18,20,16)(15,19);(1,2)(3,4)(5,7)(6,10,12,17)(8,13,15,20)(9,14)(11,16)(18,19);(2,4)(5,8)(6,11)(7,13,14,20)(9,15)(10,16,17,18)(12,19);(5,7)(6,10,12,17)(8,13,15,20)(9,14)(11,18)(16,19)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225668, #72225669, #72225677, … |
| maximal normal subgroup of | #72225668, #72225669, #72225677, … |
| maximal quotient of | #72265892 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367292);