Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2398 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18)(19,21)(20,22)(23,24);(9,10)(11,14)(12,13)(15,16)(18,21)(19,24)(22,23);(1,3,5,4)(2,6,7,8)(10,13)(14,16);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(18,22)(21,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224785, #72311797 |
| maximal normal subgroup of | #72224785, #72311797 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402813);