Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1894 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)(5,6,9,10)(7,11,12,8)(15,19)(18,20)(21,22)(23,29)(24,30)(25,33)(26,27)(28,36)(31,32)(34,35);(6,10)(7,12)(14,17)(18,20)(22,27)(23,30)(28,33)(31,34);(5,7,9,12)(6,11,10,8)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,34)(26,30)(27,29)(28,35)(31,36)(32,33);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15,16,19)(14,18,17,20)(21,24)(22,23)(25,35)(26,29)(27,30)(28,34)(31,33)(32,36);(5,8,9,11)(6,7,10,12)(21,25)(22,28)(23,31)(24,32)(26,36)(27,33)(29,35)(30,34)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3248806);