Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,24,21)(25,26,28,29)(27,30,31,32);(2,6)(4,8)(9,10,12,14)(11,13,16,15)(17,19,20,24)(18,21,22,23)(26,29)(27,31);(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27,28,31)(26,30,29,32);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,32)(28,29)(30,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232585, #72232593, #72232640, … |
| maximal normal subgroup of | #72232585, #72232593, #72232640, … |
| maximal quotient of | #72328538, #72331450 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2751306);