Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2349 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,22)(21,24)(25,26,29,32)(27,28,31,30);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(17,19,22,24)(18,21,20,23)(25,27,29,31)(26,30,32,28);(3,7)(5,8)(10,15)(12,14)(18,20)(19,24)(25,28,29,30)(26,31,32,27);(1,3)(2,5)(4,7)(6,8)(9,12,13,14)(10,11,15,16)(17,20,22,18)(19,21,24,23)(25,28)(26,27)(29,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,24)(22,23)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=g,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 | #72266510, #72266517, #72266523, … |
| maximal normal subgroup of | #72266510, #72266517, #72266523, … |
| maximal quotient of | #72334854 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1814809);