Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,11)(7,8,12,10)(13,14,18,17)(15,19,20,16)(21,22,26,30)(23,28,24,27)(25,33,34,29)(31,35,32,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,23,26,24)(22,27,30,28)(25,32,34,31)(29,36,33,35);(5,6,9,11)(7,10,12,8)(13,16,18,19)(14,15,17,20)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,17,18,14)(15,19,20,16)(21,22)(23,27)(24,28)(25,33)(26,30)(29,34)(31,36)(32,35);(5,8,9,10)(6,7,11,12)(13,15)(14,19)(16,17)(18,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5538104);