Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436971 |
| 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 | |
975 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)(3,11,5,22)(4,17)(6,12)(7,23)(8,13)(9,15,25,16)(10,20)(14,24,18,32)(19,27)(21,28)(26,29,31,30);(1,3,7,16,8,5,6,15)(2,9,12,11,13,25,23,22)(4,18,20,29,21,14,19,30)(10,24,17,31,27,32,28,26);(1,4)(2,10)(3,14)(5,18)(6,20)(7,19)(8,21)(9,24)(11,26)(12,28)(13,27)(15,30)(16,29)(17,23)(22,31)(25,32);(1,5,8,3)(2,11)(4,18,21,14)(6,15,7,16)(9,12)(10,26)(13,22)(17,32)(19,29,20,30)(23,25)(24,28)(27,31)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=efi,e^2=i,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,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 | #72237066, #72237078, #72237082, … |
| maximal normal subgroup of | #72237066, #72237078, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344168);