Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31261 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24294 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#444 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
573 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)(4,20)(5,12)(6,10)(7,21)(8,13)(11,23)(14,30)(15,32)(16,29)(17,25)(18,26)(19,28)(22,24)(27,31);(1,3)(2,9,12,28)(4,15,21,30)(5,19)(6,18)(7,14)(8,16)(10,26,13,29)(11,17)(20,32)(22,27,23,25)(24,31);(2,10)(3,14)(6,8)(7,11)(9,25)(12,13)(15,26)(16,31)(17,18)(19,32)(20,24)(22,23)(27,28)(29,30);(1,4)(2,11)(3,15)(5,21)(6,23)(7,10)(8,22)(9,17)(12,24)(13,20)(14,26)(16,27)(18,25)(19,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=fh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417277);