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 | |
2703 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)(4,6)(7,8)(9,10,13,15)(11,16,12,14)(17,18,20,21)(19,23,24,22)(25,26,29,32)(27,31,28,30);(2,3)(6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,22)(20,21)(23,24)(27,28)(30,31);(4,8)(6,7)(10,15)(14,16)(18,21)(22,23)(25,27,29,28)(26,30,32,31);(1,3,5,2)(4,6,8,7)(9,12,13,11)(10,14,15,16)(17,19,20,24)(18,22,21,23)(25,28,29,27)(26,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fghi,d^2=ghi,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.
| maximal subgroup of | #72326557, #72326558, #72326657 |
| maximal normal subgroup of | #72326557, #72326558, #72326657 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3444813);