Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| 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 | C2×#2407 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,15)(12,16)(13,14)(17,18)(19,20)(21,22)(23,24);(6,8)(10,14)(12,15)(18,22)(20,23);(1,2,4,3)(5,6)(7,8)(17,19)(18,23)(20,22)(21,24);(1,3,4,2)(9,11,13,16)(10,15,14,12);(9,12,13,15)(10,16,14,11)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^9|c^2=i,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72261124, #72261286, #72261760, … |
| maximal normal subgroup of | #72261124, #72261286, #72261760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 570877);