Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3498 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(1,2)(3,4)(8,16)(11,18)(14,20)(17,19);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4)(8,16)(11,18)(14,20)(17,19);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285985, #72286029, #72286058, … |
| maximal normal subgroup of | #72285985, #72286029, #72286058, … |
| maximal quotient of | #72306555, #72306648, C2×#6379015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286211);