Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
5718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(6,8)(13,14,17,16)(15,19,20,18);(2,4)(10,12)(14,16)(18,19);(1,2,3,4)(9,10)(11,12)(14,16)(18,19);(1,2,3,4)(5,6)(7,8)(13,15,17,20)(14,18,16,19);(1,2)(3,4)(13,16,17,14)(15,18,20,19)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232138, #72232150, #72232154, … |
| maximal normal subgroup of | #72232138, #72232150, #72232154, … |
| maximal quotient of | #72278393, #72279012, #72299585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3765874);