Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3001 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)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,23,24,22)(26,28)(30,31);(4,8)(6,7)(18,20)(19,24)(25,26,29,28)(27,31,32,30);(2,3)(6,7)(18,20)(22,23)(26,28)(27,32);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27,29,32)(26,30,28,31);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72328090, #72328116, #72328215 |
| maximal normal subgroup of | #72328090, #72328116, #72328215 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5075964);