Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17242 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,7)(4,8)(6,9,12,11)(13,22,28,19)(14,21)(15,25)(16,24)(17,26)(18,23)(20,27)(29,32,30,31);(1,3,12,27,5,14,6,15)(2,7,11,20,10,21,9,25)(4,13,18,30,16,28,17,29)(8,22,23,31,24,19,26,32);(1,4)(2,8)(3,13,14,28)(5,16)(6,18)(7,19,21,22)(9,23)(10,24)(11,26)(12,17)(15,30,27,29)(20,31,25,32);(2,9)(3,14)(4,16)(6,12)(7,20)(8,23)(10,11)(19,31)(21,25)(22,32)(24,26)(29,30)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386975);