Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,7)(6,9)(8,12)(10,11)(13,14,18,16)(15,20,17,19)(22,26)(24,32)(27,29)(30,36);(2,3)(6,8)(7,11)(13,15,18,17)(14,19,16,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,3,4,2)(5,8,10,6)(7,12,11,9)(13,16,18,14)(15,19,17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,10,12)(6,11,8,7)(13,17,18,15)(14,19,16,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4887484);