Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #2539 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1665 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)(11,14)(12,13)(15,16)(17,18,22,19,21,25,31,28)(20,23,30,26,29,24,32,27);(1,3)(2,5)(4,8)(6,7)(17,19,22,25,21,28,31,18)(20,27,30,23,29,26,32,24);(3,8)(5,7)(10,13)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,25)(27,28)(31,32);(2,6)(5,7)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32);(9,11,12,15)(10,14,13,16)(20,29)(23,24)(26,27)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284345);