Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,12)(7,8,11,10)(13,14)(15,19)(16,17)(18,20);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(21,22)(23,25)(24,26)(27,28);(5,7)(6,8)(9,11)(10,12)(13,16,17,14)(15,19,20,18);(1,2)(3,4)(5,8)(6,11)(7,12)(9,10)(13,15)(14,19)(16,18)(17,20)(21,23,24,27)(22,25,26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1208814);