Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504388 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26563 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13318 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
515 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)(3,14)(4,18)(5,22)(6,12)(7,28)(8,13)(9,10)(11,26)(15,23)(16,27)(17,21)(19,20)(24,25)(29,31)(30,32);(1,3)(2,9)(4,11)(5,23)(6,19)(7,18)(8,17)(10,26)(12,15)(13,29)(14,22)(16,21)(20,24)(25,31)(27,32)(28,30);(1,2)(3,15)(4,10)(5,24)(6,28)(7,12)(8,13)(9,19)(11,27)(14,32)(16,26)(17,30)(18,29)(20,31)(21,23)(22,25);(1,4,7,14,8,20,6,21)(2,10,12,32,13,31,28,23)(3,16,18,11,17,22,19,24)(5,15,26,29,27,30,25,9)Polycyclic presentation:
2^9|d^2=fh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ehi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411584);