Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2698 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)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,32)(28,31)(29,30);(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,20,24)(26,29)(31,32);(2,6)(3,8)(11,16)(13,14)(18,20)(23,24)(25,27,30,28)(26,31,29,32);(1,3,5,8)(2,7,6,4)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,24,21,23)(25,28,30,27)(26,32,29,31);(1,4,5,7)(2,8,6,3)(9,10)(11,14)(12,15)(13,16)(17,21,22,19)(18,24,20,23)(25,29,30,26)(27,32,28,31)Polycyclic presentation:
2^9|b^2=fhi,c^2=fh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72303510, #72303512, #72303520, … |
| maximal normal subgroup of | #72303510, #72303512, #72303520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4858596);