Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1957 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)(5,6,8,10)(7,9,12,11)(15,19)(18,20)(21,22,27,31)(23,29,24,28)(25,36,30,26)(32,33,34,35);(5,6)(7,11)(8,10)(9,12)(14,17)(18,20)(21,23,27,24)(22,28,31,29)(25,32,30,34)(26,33,36,35);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,29,31,28)(25,32,30,34)(26,33,36,35);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,25,27,30)(22,26,31,36)(23,32,24,34)(28,33,29,35);(5,7)(6,9)(8,12)(10,11)(13,15,16,19)(14,18,17,20)(21,26)(22,30)(23,33)(24,35)(25,31)(27,36)(28,34)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5416798);