Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,8)(13,14,17,16)(15,19,20,18);(9,10,11,12)(14,16)(18,19);(5,6)(7,8)(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)(13,15,17,20)(14,18,16,19);(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72252068, #72260984, #72261014, … |
| maximal normal subgroup of | #72252068, #72260984, #72261014, … |
| maximal quotient of | #72252077, #72298364, #72298367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 555555);