Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4004 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)(9,10,13,12)(11,15,16,14)(18,22)(19,23);(6,8)(10,12)(14,15)(17,18)(19,20)(21,22)(23,24);(5,6,7,8)(11,16)(14,15)(18,22)(19,23);(1,2)(3,4)(9,11,13,16)(10,14,12,15)(17,19,21,23)(18,20,22,24);(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=fgh,c^2=g,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232267, #72232269, #72232273 |
| maximal normal subgroup of | #72232267, #72232269, #72232273 |
| maximal quotient of | #72330567, #72331037 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 600317);