Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
7998 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)(13,14,17,18)(15,19,20,16);(10,12)(14,18)(16,19);(5,6)(7,8)(9,10,11,12)(14,18)(15,20);(1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,20)(18,19);(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=f,c^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240152, #72240155, #72240165, … |
| maximal normal subgroup of | #72240152, #72240155, #72240165, … |
| maximal quotient of | #72250286, #72252575, #72298114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 594781);