Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2491 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,16,12)(18,22)(19,23)(25,26)(27,31)(28,29)(30,32);(1,2,4,5)(3,7,8,6)(9,11,13,16)(10,12,14,15)(17,18,21,22)(19,20,23,24)(26,28)(27,32);(2,5)(6,7)(10,14)(12,15)(18,22)(19,23)(25,27,29,32)(26,30,28,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fgi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303423, #72303510, #72303515, … |
| maximal normal subgroup of | #72303423, #72303510, #72303515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1607347);