Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,9,11)(7,8,12,10)(14,18)(15,20)(22,26)(24,27);(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,16,20,19)(21,22,25,26)(23,27,28,24);(6,11)(8,10)(13,15,17,20)(14,19,18,16)(21,23,25,28)(22,24,26,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,20);(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72331795, #72331799, #72331918 |
| maximal normal subgroup of | #72331795, #72331799, #72331918 |
| maximal quotient of | #72333162 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 914654);