Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #438346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1158 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)(3,11)(4,15)(5,19)(6,10)(7,13)(8,18)(9,21)(12,24)(14,26)(16,22)(17,25)(20,23)(27,28)(29,31)(30,32);(1,3,5,20,6,14,21,13)(2,7,19,11,10,23,9,26)(4,16,17,12,18,30,31,27)(8,24,29,32,15,28,25,22);(1,2)(3,7,14,23)(4,8)(5,9)(6,10)(11,20,26,13)(12,22,27,32)(15,18)(16,24,30,28)(17,25)(19,21)(29,31);(1,4,5,17,6,18,21,31)(2,8,9,25,10,15,19,29)(3,12,13,16,14,27,20,30)(7,22,11,24,23,32,26,28)Polycyclic presentation:
2^9|b^2=eg,c^2=f,d^2=e,e^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,g^b=gi,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, 345542);