Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,16)(11,15,12,14)(17,18)(19,23)(20,24)(21,22);(6,8)(9,10,13,16)(11,14,12,15)(17,19)(18,20)(21,23)(22,24);(5,6)(7,8)(9,11,13,12)(10,14,16,15)(17,20,22,23)(18,19,21,24);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,21,22,18)(19,23,24,20);(9,12,13,11)(10,15,16,14)(17,19)(18,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2667940);