Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C43 |
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 | |
6421 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)(15,17)(16,18)(19,20);(1,2)(3,4)(9,10,11,12)(14,15)(18,19);(1,2,3,4)(9,10)(11,12)(16,20)(18,19);(5,6,7,8)(13,15,17,14)(16,19,20,18);(1,2)(3,4)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231349, #72232154, #72232163, … |
| maximal normal subgroup of | #72231349, #72232154, #72232163, … |
| maximal quotient of | #72278012, #72278393, C2×#2306470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2213666);