Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
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 | |
5552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(6,8)(13,14,16,18)(15,19,20,17);(2,3)(10,12)(13,14,16,18)(15,17,20,19);(9,10,11,12)(13,14)(15,17)(16,18)(19,20);(2,3)(5,6,7,8)(13,14)(15,17)(16,18)(19,20);(1,3,4,2)(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232232, #72232245, #72232467, … |
| maximal normal subgroup of | #72232232, #72232245, #72232467, … |
| maximal quotient of | #72330566, #72330647, #72331568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2830478);