Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| 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 | |
2075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,12)(11,15)(17,18,21,22)(19,20,23,24)(25,26)(27,29)(28,31)(30,32);(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,19,21,23)(18,20,22,24)(26,27)(30,31);(3,7)(6,8)(9,10,13,12)(11,14,15,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,26)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,18)(19,20)(21,22)(23,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=fh,c^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2148192);