Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427835 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,11)(4,14)(5,17)(7,21)(9,18)(12,27)(13,19)(22,32)(23,25)(24,26)(28,29)(30,31);(1,2)(3,7)(4,15)(5,18)(6,10)(8,14)(9,17)(11,25)(12,22)(13,23)(16,24)(19,21)(20,26)(27,32)(28,31)(29,30);(1,3)(2,7)(4,12)(5,19)(6,13)(8,22)(9,25)(10,23)(11,17)(14,29)(15,30)(16,27)(18,21)(20,28)(24,31)(26,32);(1,4,5,20,6,14,17,16)(2,8,9,26,10,15,18,24)(3,12,19,28,13,29,11,27)(7,22,25,32,23,30,21,31)Polycyclic presentation:
2^9|d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335031);