Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×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 | |
5057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,20)(21,23)(22,24);(10,14)(12,15)(17,19,21,24)(18,22,23,20);(5,6,7,8)(9,11,13,16)(10,15,14,12)(17,20,21,22)(18,19,23,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24);(9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=h,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232218, #72232240, #72232243, … |
| maximal normal subgroup of | #72232218, #72232240, #72232243, … |
| maximal quotient of | #72330020, #72331442, #72332749 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4840197);