Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196350 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #34981 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,11,15,18)(9,12)(13,16,19,20)(14,17);(2,4)(5,7)(6,10)(8,13,15,19)(9,14)(11,16,18,20)(12,17);(5,8)(6,11,12,18)(7,13,14,19)(9,15)(10,16)(17,20);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225625, #72225628, … |
| maximal normal subgroup of | #72225620, #72225625, #72225628, … |
| maximal quotient of | #72255656, #72265630, #72265635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275980);