Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,12)(7,11,8,10)(15,19)(17,20)(21,22)(23,28)(24,30)(25,33)(26,27)(29,34)(31,32)(35,36);(2,4)(7,8)(10,11)(13,14,16,18)(15,17,19,20)(22,27)(23,30)(25,34)(32,35);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,19)(18,20)(21,23,26,30)(22,28,27,24)(25,31,34,36)(29,35,33,32);(5,8,9,7)(6,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,24,26,28)(22,23,27,30)(25,35,34,32)(29,36,33,31);(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,33)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=h,d^2=i,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, 3068320);