Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
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 | |
3506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,11)(14,15)(17,18)(19,23)(20,21)(22,24)(26,27)(28,32);(4,8)(6,7)(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,23,24,21)(25,26,29,27)(28,31,32,30);(2,3)(6,7)(10,11)(14,15)(18,24)(19,20)(25,26,29,27)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,14,16,15)(17,20,22,19)(18,23,24,21)(25,27,29,26)(28,30,32,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,23)(18,19,24,20)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|b^2=fghi,c^2=fgi,d^2=i,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.
| maximal subgroup of | #72234029, #72234034, #72234047, … |
| maximal normal subgroup of | #72234029, #72234034, #72234047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5815593);