Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #635984 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2856 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4582 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)(6,7)(9,10,13,15)(11,16,12,14)(19,23)(22,24);(9,11,13,12)(10,14,15,16)(18,21)(22,24);(5,6,8,7)(10,15)(14,16)(17,18)(19,22)(20,21)(23,24);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24);(9,12,13,11)(10,16,15,14)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72261501, #72261504, #72261535 |
| maximal normal subgroup of | #72261501, #72261504, #72261535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 548546);