Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2181 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)(7,8,10,12)(13,14,16,18)(15,17,19,20)(22,23)(24,31)(25,33)(35,36);(5,6)(7,12)(8,10)(9,11)(15,19)(17,20)(21,22,26,23)(24,29,31,27)(25,28,33,30)(32,36,34,35);(5,7,9,10)(6,8,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,36,34,35);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,31)(22,27,23,29)(25,34,33,32)(28,36,30,35);(5,8)(6,10)(7,11)(9,12)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2508813);