Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2831 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18,21,23)(19,20,24,22)(25,26)(27,32)(28,29)(30,31);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,23,20)(26,31)(27,28);(3,7)(6,8)(10,15)(12,14)(18,23)(19,24)(25,27,30,28)(26,32,31,29);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,28,30,27)(26,29,31,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,15)(13,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3492841);