Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428865 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15391 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9,18,27)(5,19)(6,10)(7,11)(8,14)(13,29,30,25)(15,23)(16,32,31,22)(17,28,20,26)(21,24);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,19,23)(4,16,17,30,18,31,20,13)(9,25,26,32,27,29,28,22);(1,2)(3,12)(4,9)(5,10)(6,19)(7,11)(8,21)(13,29)(14,24)(15,23)(16,22)(17,26)(18,27)(20,28)(25,30)(31,32);(1,4,5,20,7,18,6,17)(2,9,10,28,11,27,19,26)(3,13,14,31,15,30,21,16)(8,22,12,29,24,32,23,25)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338910, #72338924, #72338926 |
| maximal normal subgroup of | #72338910, #72338924, #72338926 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336061);