Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | #52117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,25,26,32)(19,28,20,24,29,23,30,31);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,22)(21,29)(24,32)(25,28)(26,30)(27,31);(10,14)(13,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,27)(25,31)(28,32);(2,5)(6,8)(18,25)(19,29)(20,30)(27,32);(1,3,4,7)(2,6,5,8)(17,18,26,32,21,25,22,27)(19,28,30,31,29,23,20,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,e^b=ei,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6309559);