Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #34715 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,23,27,22,26,28,21)(19,24,30,25,29,32,20,31);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,28)(21,31)(22,29)(23,30)(25,27)(26,32);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,20,22,30)(18,25,26,31)(19,28,29,23)(21,24,27,32);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,21,28,26,22,27,23,18)(19,25,20,24,29,31,30,32);(10,14)(13,15)(18,26)(19,29)(20,30)(21,27)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,d^b=dfi,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285013, #72285065, #72285173, … |
| maximal normal subgroup of | #72285013, #72285065, #72285173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6332821);