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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2240 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)(5,6,8,10)(7,11,12,9)(13,14,17,16)(15,18,20,19)(21,22)(23,28)(24,25)(26,27)(29,34)(30,35)(31,36)(32,33);(7,12)(9,11)(13,14,17,16)(15,19,20,18)(22,27)(23,30)(25,29)(31,33);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19)(21,23,26,30)(22,28,27,35)(24,33,34,31)(25,32,29,36);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(21,24)(22,29)(23,31)(25,27)(26,34)(28,32)(30,33)(35,36);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,25,26,29)(22,24,27,34)(23,32,30,36)(28,33,35,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=h,d^2=fgi,e^2=h,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, 1654302);