Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
2385 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)(5,6,9,8)(7,11,12,10)(13,14,17,19)(15,18,16,20)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,32,35,36);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,18)(17,19)(21,23,25,30)(22,26,28,33)(24,29,31,35)(27,32,34,36);(13,15,17,16)(14,18,19,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,17,15)(14,18,19,20)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,35)(34,36);(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,19)(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,32,33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4770053);