Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26633 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,24,30,6,14,23,31,8,13,25,20)(3,9,16,28,26,4,18,19,10,27,21,22,17,15,29,32);(1,2)(3,15)(4,10)(5,20)(6,14)(7,13)(8,12)(9,17)(11,25)(16,22)(18,19)(21,28)(23,30)(24,31)(26,27)(29,32);(1,3)(2,9)(4,12)(5,21)(6,10)(7,26)(8,17)(11,22)(13,15)(14,27)(16,23)(18,25)(19,20)(24,29)(28,31)(30,32);(1,4,7,9,6,15,8,27)(2,10,13,17,14,3,12,26)(5,22,24,19,23,28,25,32)(11,29,20,16,31,18,30,21)Polycyclic presentation:
2^9|a^2=e,d^2=fh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,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, 411676);