Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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:
(5,6,9,10)(7,8,12,11)(15,19)(17,20)(21,22)(23,28)(24,25)(26,27)(29,34)(30,36)(31,33)(32,35);(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,17,19,20)(22,27)(25,29)(28,35)(31,36);(1,2,3,4)(6,10)(7,12)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,36)(30,34);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,24)(22,29)(23,30)(25,27)(26,34)(28,31)(32,33)(35,36);(5,8)(6,7)(9,11)(10,12)(21,25,26,29)(22,24,27,34)(23,31,32,36)(28,33,35,30)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1482138);