Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #376489 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14181 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 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)(3,7,12,20)(4,9,15,24)(5,8)(6,10)(11,17)(13,27)(14,21,26,18)(16,22,29,23)(19,25)(28,30)(31,32);(1,3,5,14,6,12,17,26)(2,7,11,18,10,20,8,21)(4,13,16,28,15,25,29,32)(9,19,23,30,24,27,22,31);(2,8)(3,12)(5,17)(7,18)(9,22)(10,11)(13,25)(16,29)(19,30)(20,21)(23,24)(27,31);(1,4)(2,9,10,24)(3,13)(5,16)(6,15)(7,19,20,27)(8,22,11,23)(12,25)(14,28)(17,29)(18,30,21,31)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=eh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319525 |
| maximal normal subgroup of | #72319525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283685);