Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2157 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)(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(23,29)(24,32)(26,30)(27,34);(6,7)(8,12)(14,17)(15,19)(21,22,25,28)(23,30,29,26)(24,27,32,34)(31,36,35,33);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,23,25,29)(22,26,28,30)(24,31,32,35)(27,33,34,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,14)(15,20)(16,17)(18,19)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(5,8)(6,10)(7,11)(9,12)(13,15,16,19)(14,18,17,20)(21,24)(22,27)(23,31)(25,32)(26,33)(28,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=h,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2992731);