Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,9)(8,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,29)(25,30)(27,32)(28,33)(31,34)(35,36);(2,4)(5,6,10,12)(7,11,9,8)(13,15,18,17)(14,19,16,20)(22,23)(25,33)(27,29)(31,36);(1,2,3,4)(5,7,10,9)(6,11,12,8)(13,16,18,14)(15,19,17,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,8)(6,7)(9,12)(10,11)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,9,10,7)(6,8,12,11)(13,17,18,15)(14,19,16,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|b^2=fg,c^2=fhi,d^2=f,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, 1052771);