Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4686 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)(9,10,13,12)(11,15,16,14)(18,22)(20,24);(6,7)(10,12)(14,15)(17,18,21,22)(19,20,23,24);(5,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,23)(18,20,22,24);(1,2)(3,4)(5,7,8,6)(9,12,13,10)(11,15,16,14);(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fg,d^2=g,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 | #72328105, #72328167, #72328189, … |
| maximal normal subgroup of | #72328105, #72328167, #72328189, … |
| maximal quotient of | #72330255, #72333143 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 753776);