Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,13)(11,15)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32);(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,23,20)(26,29)(27,31);(11,15)(14,16)(18,23)(20,22)(25,26,28,29)(27,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,19,23,24)(25,26)(27,32)(28,29)(30,31);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fh,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3140846);