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) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4086 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)(9,10)(11,15)(12,14)(13,16)(18,22)(19,24);(5,6,7,8)(9,11,14,13)(10,15,12,16)(17,18,21,22)(19,23,24,20);(5,6)(7,8)(9,10,14,12)(11,16,13,15)(17,19)(18,23)(20,22)(21,24);(1,2)(3,4)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,22)(21,24);(9,13,14,11)(10,15,12,16)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|b^2=fh,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 | #72232417, #72232439, #72232460, … |
| maximal normal subgroup of | #72232417, #72232439, #72232460, … |
| maximal quotient of | #72330085, #72330305, #72331359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3768062);