Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #2416 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,25,22,26,30,32)(19,23,24,31,27,20,28,29);(2,5)(3,7);(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,30)(29,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(19,27)(20,23)(24,28)(29,31);(2,5)(6,8)(17,20,21,29,22,23,30,31)(18,24,25,27,26,28,32,19)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=df,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239435, #72256894, #72256909, … |
| maximal normal subgroup of | #72239435, #72256894, #72256909, … |
| maximal quotient of | #72239393, #72257830 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6313717);