Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #37672 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,11)(7,9,12,10)(14,17)(15,19)(18,20);(1,2)(3,4)(6,9)(7,12)(10,11)(13,14)(15,20)(16,18)(17,19);(2,4)(15,19)(17,20);(6,9)(7,12)(10,11)(13,14)(15,20)(16,18)(17,19)Polycyclic presentation:
2^9|a^2=hi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^b=fi,e^c=ei,e^d=eh,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72311631, #72311653, #72311678, … |
| maximal normal subgroup of | #72311631, #72311653, #72311678, … |
| maximal quotient of | C2×#246012 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153208);