Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14921 |
| 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 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,19)(3,8,20,23,13,24,14,26)(4,15,21,25,17,30,18,9)(12,27,31,32,16,29,28,22);(1,3)(2,8)(4,16,17,12)(5,13)(6,20)(7,14)(9,22,25,32)(10,26)(11,24)(15,29,30,27)(18,28,21,31)(19,23);(1,4,5,17)(2,9)(3,12,13,16)(6,18,7,21)(8,22)(10,15)(11,25)(14,28,20,31)(19,30)(23,27)(24,32)(26,29);(6,7)(10,19)(14,20)(15,30)(18,21)(23,26)(27,29)(28,31)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313142);