Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,23)(22,24);(6,8)(9,11,12,16)(10,13,14,15)(17,19,21,24)(18,22,23,20);(5,6,7,8)(10,14)(13,15)(19,24)(20,22);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24);(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232642, #72232657, #72232709, … |
| maximal normal subgroup of | #72232642, #72232657, #72232709, … |
| maximal quotient of | #72328589, #72328783, #72329935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1008159);