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 | |
5322 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)(15,19)(16,18)(17,20);(2,3)(9,10,11,12)(13,15,18,17)(14,19,16,20);(2,3)(9,10)(11,12)(13,14,18,16)(15,20,17,19);(2,3)(5,6)(7,8)(13,16,18,14)(15,19,17,20);(1,3,4,2)(13,17,18,15)(14,19,16,20)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fg,d^2=fg,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 | #72297514, #72297532, #72297553, … |
| maximal normal subgroup of | #72297514, #72297532, #72297553, … |
| maximal quotient of | #72298251, #72298256, #72298354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3768061);