Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441632 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15688 |
| 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 | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,12)(4,15)(5,9)(6,10,16,23)(8,18)(13,25,31,30)(14,28)(17,24,19,32)(20,26)(21,27)(22,29);(1,3,6,4,7,14,16,18)(2,8,23,28,11,15,10,12)(5,19,20,31,21,17,22,13)(9,25,29,24,27,30,26,32);(1,4,16,14,7,18,6,3)(2,8,10,28,11,15,23,12)(5,17,20,31,21,19,22,13)(9,24,29,25,27,32,26,30);(1,5)(2,9)(3,13)(4,17)(6,22)(7,21)(8,24)(10,29)(11,27)(12,30)(14,31)(15,32)(16,20)(18,19)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348828);