Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442123 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15701 |
| 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 | |
877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,20,28,19,29)(4,21,23,31,17,30,24,10)(5,25,26,32,22,11,16,27);(1,3,8,18,7,20,6,19)(2,9,12,28,14,29,13,15)(4,22,23,16,17,5,24,26)(10,25,21,32,31,11,30,27);(1,4,7,17)(2,10)(3,16,20,26)(5,18,22,19)(6,23,8,24)(9,27)(11,28)(12,30)(13,21)(14,31)(15,25)(29,32);(1,5,8,26,7,22,6,16)(2,11,13,32,14,25,12,27)(3,17,19,23,20,4,18,24)(9,10,28,21,29,31,15,30)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349320);