Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238259 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,20,18,21)(22,24);(9,10)(11,18,19,14)(12,15)(13,16)(17,20,23,21)(22,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,21)(13,19)(15,23)(16,18)(17,24)(20,22);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,g^b=gh,d^c=di,e^c=eh,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232921, #72233068, #72233076 |
| maximal normal subgroup of | #72232921, #72233068, #72233076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145455);