Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,21,22,25,19,26)(20,28,31,29,30,24,27,32);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,21,25,26)(20,27,30,31)(24,32,28,29);(10,14)(11,15)(17,20)(18,24)(19,27)(21,32)(22,30)(23,31)(25,28)(26,29);(3,8)(5,7)(10,14)(13,16)(17,21,23,25,22,26,19,18)(20,29,31,24,30,32,27,28);(2,6)(5,7)(20,30)(24,28)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72281643, #72281760 |
| maximal normal subgroup of | #72281643, #72281760 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6329262);