Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485588 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18152 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
686 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)(3,9)(4,11)(5,18)(6,23)(7,12)(8,14)(10,21)(13,30)(15,26)(16,28)(17,29)(19,27)(20,24)(22,32)(25,31);(1,3)(2,8)(4,13)(5,19)(6,16)(7,15)(9,18)(10,28)(11,27)(12,24)(14,21)(17,20)(22,31)(23,25)(26,32)(29,30);(1,3)(2,9)(4,16)(5,20)(6,13)(7,15)(8,18)(10,27)(11,28)(12,26)(14,22)(17,19)(21,31)(23,30)(24,32)(25,29);(1,4,7,6)(3,13,15,16)(5,21,22,17)(9,26)(10,29)(14,31,20,19)(18,32)(28,30)Polycyclic presentation:
2^9|d^2=h,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,e^d=ei,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392784);