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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1832 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)(5,6,9,11)(7,12,10,8)(21,22)(23,30)(24,28)(25,29)(26,27)(31,33)(32,36)(34,35);(5,7,9,10)(6,8,11,12)(13,14,17,16)(15,19,20,18)(22,27)(23,28)(29,35)(31,36);(7,10)(8,12)(13,15,17,20)(14,18,16,19)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,35)(34,36);(1,2)(3,4)(5,8,9,12)(6,10,11,7)(21,24)(22,28)(23,27)(25,33)(26,30)(29,31)(32,34)(35,36);(5,6)(7,8)(9,11)(10,12)(13,16,17,14)(15,19,20,18)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fh,d^2=fg,e^2=fh,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, 2672532);