Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59126 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(10,13)(11,15)(18,21)(19,23)(22,24);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(18,22)(21,24);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,24,20,22,19,21);(2,5)(6,8)(10,13)(14,16)(18,22)(21,24);(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^9|c^2=fgh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72316734, #72316927, #72316949 |
| maximal normal subgroup of | #72316734, #72316927, #72316949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102089);