Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
2034 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)(5,6,10,12)(7,8,11,9)(14,18)(15,16)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,36)(34,35);(5,7,10,11)(6,8,12,9)(14,18)(19,20)(22,27)(23,28)(29,35)(30,36);(5,8,10,9)(6,7,12,11)(13,14,17,18)(15,19,16,20)(21,23,26,28)(22,24,27,32)(25,30,34,36)(29,31,35,33);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,16)(14,19,18,20)(21,24)(22,28)(23,27)(25,33)(26,32)(29,30)(31,34)(35,36);(5,9,10,8)(6,11,12,7)(13,16,17,15)(14,20,18,19)(21,25)(22,29)(23,30)(24,31)(26,34)(27,35)(28,36)(32,33)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1170308);