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 | |
3254 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)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32);(9,10)(11,13)(12,14)(15,16)(17,19,22,20)(18,23,25,26)(21,28,30,27)(24,32,29,31);(6,8)(9,10,12,14)(11,13,16,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30);(9,11)(10,13)(12,16)(14,15)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fh,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 | #72266869, #72266872, #72267279, … |
| maximal normal subgroup of | #72266869, #72266872, #72267279, … |
| maximal quotient of | #72266899, #72267289, #72267425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4840679);