Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,12)(7,8,11,10)(13,14,18,19)(15,17,20,16)(22,25)(23,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,19)(18,20);(1,2)(3,4)(13,16,18,17)(14,15,19,20)(21,22,24,25)(23,26,27,28);(5,7)(6,8)(9,11)(10,12)(13,17,18,16)(14,15,19,20)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,8)(6,11)(7,12)(9,10)(13,15)(14,17)(16,19)(18,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^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, 1194382);