Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2228 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)(6,8)(7,11)(14,17)(15,19)(21,22,26,29)(23,30,33,27)(24,34,28,25)(31,36,35,32);(14,17)(18,20)(21,23,26,33)(22,27,29,30)(24,31,28,35)(25,36,34,32);(5,6,9,8)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,35)(25,32)(26,33)(28,31)(29,30)(34,36);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(21,24,26,28)(22,25,29,34)(23,31,33,35)(27,36,30,32);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,32)(24,29)(26,34)(27,31)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1102984);