Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(5,6,7,8)(11,15)(13,16)(17,18,21,22)(19,25,26,23)(20,28,30,24)(27,31,29,32);(9,10,12,14)(11,16,15,13)(18,22)(19,26)(24,28)(27,29);(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31);(1,2)(3,4)(5,6)(7,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32);(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72268294, #72268296, #72268312, … |
| maximal normal subgroup of | #72268294, #72268296, #72268312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 671437);