Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| 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 | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5426 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)(6,8)(10,12)(11,15)(17,18)(19,20)(21,22)(23,24);(9,10,13,12)(11,14,15,16)(18,22)(19,23);(5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,23)(18,20,22,24);(1,2)(3,4)(9,11)(10,14)(12,16)(13,15);(9,12,13,10)(11,16,15,14)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72232088, #72232127, #72232141, … |
| maximal normal subgroup of | #72232088, #72232127, #72232141, … |
| maximal quotient of | C2×#613897, #72330392, #72336227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 521093);