Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 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)(3,12,16,22)(4,8)(5,9)(6,11)(7,15,21,14)(10,19)(13,28,31,27)(17,24)(18,25)(20,29,23,30)(26,32);(1,3)(2,7)(4,13)(5,14)(6,16)(8,23)(9,22)(10,12)(11,21)(15,19)(17,29)(18,31)(20,25)(24,27)(26,28)(30,32);(1,4)(2,8)(3,13,16,31)(5,17)(6,18)(7,20,21,23)(9,24)(10,26)(11,25)(12,27,22,28)(14,29,15,30)(19,32);(2,9)(3,14)(5,19)(7,21)(8,24)(10,11)(13,29)(15,16)(17,32)(20,23)(25,26)(30,31)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,e^d=ei,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319564 |
| maximal normal subgroup of | #72319564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331787);