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) = C26×C4 |
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 | |
3473 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)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,24)(21,22)(25,26,29,30)(27,31,32,28);(2,5)(6,7)(9,11,13,16)(10,14,15,12)(17,19,21,24)(18,20,22,23)(26,30)(28,31);(3,8)(6,7)(10,15)(11,16)(18,22)(20,23)(25,27,29,32)(26,31,30,28);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,31)(28,30)(29,32);(1,2)(3,7)(4,5)(6,8)(9,12)(10,16)(11,15)(13,14)(17,19)(18,23)(20,22)(21,24)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fhi,e^2=fhi,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 | #72278957, #72300707, #72300749, … |
| maximal normal subgroup of | #72278957, #72300707, #72300749, … |
| maximal quotient of | #72279047, #72299575, #72299655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3163230);