Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,29)(30,32);(2,5)(6,7)(9,11,13,16)(10,14,15,12)(18,22)(20,23)(25,27)(26,30)(28,31)(29,32);(3,8)(6,7)(10,15)(12,14)(17,19,21,24)(18,23,22,20)(25,27,29,32)(26,31,28,30);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,11,15,16)(17,19)(18,23)(20,22)(21,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,23)(18,24,22,19)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fghi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72300716, #72300749, #72300760, … |
| maximal normal subgroup of | #72300716, #72300749, #72300760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3237854);