Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3019 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)(4,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21)(26,28)(30,32);(2,3)(4,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,24,23)(27,31)(30,32);(4,8)(6,7)(10,12)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26,29,28)(27,30,31,32);(1,3,5,2)(4,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,21)(20,22)(23,24)(25,27,29,31)(26,30,28,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,19)(18,21)(20,24)(22,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72263190, #72263204, #72263209, … |
| maximal normal subgroup of | #72263190, #72263204, #72263209, … |
| maximal quotient of | #72328442 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3812355);