Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,24,28,23,19,22,27)(20,29,26,21,31,25,30,32);(1,2,5,6)(3,7,4,8)(9,11,13,12)(10,14,15,16)(17,19,23,18)(20,21,31,32)(22,27,24,28)(25,26,29,30);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,24,26,23,31,22,30)(18,25,28,32,19,29,27,21);(1,4,5,3)(2,7,6,8)(17,21,24,25,23,32,22,29)(18,26,28,31,19,30,27,20)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=ef,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348978);