Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #450134 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15958 |
| 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 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,5,8,12,6,11)(3,13,16,9,18,24,17,15)(4,19,22,28,23,27,21,10)(14,29,20,26,30,25,31,32);(1,3)(2,9)(4,20,23,31)(5,24)(6,17)(7,16)(8,18)(10,26,28,32)(11,13)(12,15)(14,22,30,21)(19,25,27,29);(1,4)(2,10)(3,14)(5,19)(6,21)(7,22)(8,23)(9,25)(11,27)(12,28)(13,26)(15,29)(16,20)(17,31)(18,30)(24,32);(1,5,6,12,8,11,7,2)(3,15,16,24,18,9,17,13)(4,19,21,28,23,27,22,10)(14,29,20,32,30,25,31,26)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=e,e^2=i,f^2=i,g^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,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251785, #72251792, #72251800, … |
| maximal normal subgroup of | #72251785, #72251792, #72251800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357330);