Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
3476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,8,7)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,21)(22,23)(25,26,28,30)(27,31,32,29);(2,6)(3,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,21)(18,20,23,24)(27,32)(29,31);(2,6)(4,7)(11,16)(13,15)(18,23)(19,21)(25,27)(26,29)(28,32)(30,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,29)(28,32)(30,31);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,13,14,15)(17,21,22,19)(18,20,23,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300847, #72300860, #72300900, … |
| maximal normal subgroup of | #72300847, #72300860, #72300900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4523599);