Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15463 |
| 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 | |
1143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,8,6,19)(3,9,12,25,14,21,18,23)(4,15,16,10,17,26,20,22)(13,28,27,31,29,32,30,24);(2,8)(3,12)(5,6)(9,23)(13,27)(14,18)(15,26)(16,20)(21,25)(24,28)(29,30)(31,32);(1,3)(2,9,8,21)(4,13)(5,18)(6,12)(7,14)(10,24,22,31)(11,23,19,25)(15,28,26,32)(16,30)(17,29)(20,27);(1,4)(2,10)(3,13)(5,16)(6,20)(7,17)(8,22)(9,24)(11,26)(12,27)(14,29)(15,19)(18,30)(21,31)(23,32)(25,28)Polycyclic presentation:
2^9|a^2=g,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216142, #72216217, … |
| maximal normal subgroup of | #72215936, #72216142, #72216217, … |
| maximal quotient of | C2×#432668, #72252100, #72265597 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339864);