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) = C27 |
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 | |
3104 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)(9,10)(11,12)(13,15)(14,16)(17,18,21,24)(19,20,23,22)(25,26)(27,30)(28,29)(31,32);(2,3)(4,8)(9,11,13,16)(10,14,15,12)(17,19,21,23)(18,22,24,20)(26,28)(30,31);(2,3)(6,7)(10,15)(12,14)(19,23)(20,22)(25,27,29,32)(26,30,28,31);(1,3,5,2)(4,6,8,7)(9,12,13,14)(10,11,15,16)(17,20,21,22)(18,23,24,19)(25,27)(26,31)(28,30)(29,32);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fgh,d^2=gh,e^2=fgh,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 | #72327861, #72327883, #72327886, … |
| maximal normal subgroup of | #72327861, #72327883, #72327886, … |
| maximal quotient of | #72328521, #72330655 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3009407);