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 | |
2112 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22,26,27)(23,25,29,34)(24,28,33,35)(30,32,31,36);(2,4)(6,9)(7,12)(13,15,17,20)(14,16,18,19)(22,27)(25,34)(28,35)(32,36);(1,2,3,4)(6,9)(10,11)(14,18)(15,20)(21,23,26,29)(22,25,27,34)(24,31,33,30)(28,36,35,32);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,26,33)(22,28,27,35)(23,30,29,31)(25,32,34,36);(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,19)(18,20)(21,25)(22,29)(23,27)(24,32)(26,34)(28,31)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fghi,d^2=fg,e^2=gi,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, 1542111);