Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
2715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,8,7)(10,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,26,29,31)(27,32,28,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,23)(22,24)(25,27,29,28)(26,30,31,32);(2,6)(3,8)(11,15)(14,16)(17,19,21,23)(18,22,24,20)(25,26,29,31)(27,32,28,30);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31);(1,4,5,7)(2,3,6,8)(9,10)(11,16)(12,13)(14,15)(17,20,21,22)(18,23,24,19)(25,28,29,27)(26,30,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| Fitting subgroup of | #72260950 |
| maximal subgroup of | #72260841, #72260950 |
| maximal normal subgroup of | #72260841, #72260950 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5588699);