Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 3145728 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(19,23)(21,24)(26,29)(27,31);(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,16,15);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,21,23,24)(25,26)(27,30)(28,29)(31,32);(17,19)(18,21)(20,23)(22,24)(25,27,28,31)(26,30,29,32);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
| Fitting subgroup of | #72258826 |
| maximal subgroup of | #72258789, #72258826, #72258895, … |
| maximal normal subgroup of | #72258789, #72258826, #72258895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4323703);