Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436995 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15578 |
| 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 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,21,6,11,19,12)(3,8,5,10,14,25,20,24)(4,15,22,28,18,29,17,9)(13,27,16,26,30,32,31,23);(1,3,7,5,6,14,19,20)(2,8,12,24,11,25,21,10)(4,13,17,31,18,30,22,16)(9,26,15,32,28,23,29,27);(1,4)(2,9)(3,13)(5,16)(6,18)(7,22)(8,23)(10,27)(11,28)(12,29)(14,30)(15,21)(17,19)(20,31)(24,32)(25,26);(1,5,19,14,6,20,7,3)(2,10,21,8,11,24,12,25)(4,16,17,30,18,31,22,13)(9,27,15,23,28,32,29,26)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319859, #72320044, #72320069, … |
| maximal normal subgroup of | #72319859, #72320044, #72320069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344191);