Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(14,16)(18,20)(21,22,26,27)(23,28,29,24)(25,33,35,30)(31,32,36,34);(2,4)(5,6,9,8)(7,10,11,12)(15,19)(18,20)(22,27)(24,28)(30,33)(32,34);(1,2)(3,4)(13,14,17,16)(15,18,19,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,34)(32,33)(35,36);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,28)(22,29,27,23)(25,34,35,32)(30,31,33,36);(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,20,19,18)(21,25,26,35)(22,30,27,33)(23,31,29,36)(24,32,28,34)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2696948);