Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3660 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)(3,6)(4,5)(7,8)(10,11)(12,16)(17,18)(19,20)(21,24)(22,23)(26,29)(27,31);(2,5)(6,8)(10,11)(14,15)(17,19)(18,21)(20,23)(22,24)(27,31)(30,32);(3,7)(6,8)(9,10,13,11)(12,14,16,15)(17,20,22,21)(18,19,23,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,21,22,20)(18,19,23,24)(25,27,28,31)(26,30,29,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300712, #72300741, #72300762, … |
| maximal normal subgroup of | #72300712, #72300741, #72300762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2501913);