Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24);(10,14)(12,15)(17,18,21,19)(20,23,24,22);(1,2,4,3)(5,6)(7,8)(10,14)(11,16)(18,19)(22,23);(1,3,4,2)(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,19,21,18)(20,23,24,22);(9,12,13,15)(10,16,14,11)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fgi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72213681, #72232295, #72232589, … |
| maximal normal subgroup of | #72210448, #72213681, #72232295, … |
| maximal quotient of | #72250278, #72252083, #72293980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 446958);