Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(2,4)(13,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,16,18,19)(14,20,17,15)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,7,9,11)(6,10,8,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8,9,6)(7,12,11,10)(13,17,18,14)(15,16,20,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fi,d^2=i,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3700304);