Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,11)(8,10)(9,12)(14,16)(18,20)(21,22,26,27)(23,24,31,28)(25,33,34,29)(30,35,36,32);(2,4)(5,7,10,9)(6,11,8,12)(15,19)(18,20)(22,27)(24,28)(25,34)(30,36);(1,2)(3,4)(5,6,10,8)(7,12,9,11)(13,14,17,16)(15,18,19,20)(21,23,26,31)(22,28,27,24)(25,30,34,36)(29,35,33,32);(5,8,10,6)(7,11,9,12)(13,15,17,19)(14,18,16,20)(21,24,26,28)(22,23,27,31)(25,32,34,35)(29,30,33,36);(5,9,10,7)(6,11,8,12)(13,16,17,14)(15,20,19,18)(21,25,26,34)(22,29,27,33)(23,30,31,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1449082);