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) = C25×C4 |
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 | |
2588 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)(3,7)(4,6)(5,8)(10,14)(12,15)(18,19)(20,24)(26,29)(30,32);(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(18,19)(22,23)(27,31)(30,32);(3,8)(5,7)(10,14)(12,15)(17,18,21,19)(20,22,24,23)(25,26)(27,30)(28,29)(31,32);(1,2)(3,7)(4,6)(5,8)(9,11,13,16)(10,12,14,15)(17,19,21,18)(20,23,24,22)(25,27,28,31)(26,30,29,32);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=fhi,c^2=i,d^2=ghi,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 | #72326553, #72326596, #72326624 |
| maximal normal subgroup of | #72326553, #72326596, #72326624 |
| maximal quotient of | #72328621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3939371);