Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
7198 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)(5,6,7,8)(9,10,13,16)(11,15,12,14)(17,18,21,22)(19,20,23,24);(9,11,13,12)(10,14,16,15)(18,22)(19,23);(5,6)(7,8)(17,19)(18,20)(21,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16);(9,12,13,11)(10,15,16,14)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,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 | #72232141, #72232151, #72232170, … |
| maximal normal subgroup of | #72232141, #72232151, #72232170, … |
| maximal quotient of | C2×#613883, #72329876, #72330735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 521079);