Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
6986 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)(10,13)(11,15)(18,21)(19,23);(9,10)(11,14)(12,13)(15,16)(17,18,20,21)(19,22,23,24);(5,6)(7,8)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^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 | #72309337, #72337224, #72337227, … |
| maximal normal subgroup of | #72309337, #72337224, #72337227, … |
| maximal quotient of | #72337237, #72337364, #72339922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 520707);