Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| 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×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5208 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)(6,7)(10,13)(11,15)(17,18,22,23)(19,21,20,24);(9,10,12,13)(11,14,15,16)(18,23)(19,20);(5,6,8,7)(17,19,22,20)(18,24,23,21);(1,2)(3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,21,23,24);(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256744, #72256754, #72256773, … |
| maximal normal subgroup of | #72256744, #72256754, #72256773, … |
| maximal quotient of | #72330457, #72331325, #72332489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 524805);