Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,27)(21,29)(26,31);(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,20,24,27)(21,30,29,25)(26,32,31,28);(6,8)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32);(9,11)(10,13)(12,16)(14,15)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,28,27,32)Polycyclic presentation:
2^9|b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266752, #72266759, #72267407, … |
| maximal normal subgroup of | #72266752, #72266759, #72267407, … |
| maximal quotient of | #72266774, #72267289, #72267425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4840675);