Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | |
5116 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)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24);(6,8)(10,14)(11,15)(17,19,20,24)(18,21,23,22);(5,6)(7,8)(9,11,13,15)(10,12,14,16)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(17,18)(19,22)(20,23)(21,24);(9,12)(10,15)(11,14)(13,16)Polycyclic presentation:
2^9|b^2=g,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72328103, #72328117, #72328260, … |
| maximal normal subgroup of | #72328103, #72328117, #72328260, … |
| maximal quotient of | #72330883, #72333159, #72335835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752523);