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 | |
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:
(2,3)(7,9)(8,11)(14,17)(15,19)(21,22,25,26)(23,29,31,27)(24,28,33,34)(30,36,32,35);(5,6,10,12)(7,11,9,8)(14,17)(18,20)(22,26)(27,29)(28,34)(35,36);(1,2,4,3)(5,7,10,9)(6,11,12,8)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,32)(25,31)(26,29)(28,36)(30,33)(34,35);(5,8)(6,7)(9,12)(10,11)(21,24,25,33)(22,28,26,34)(23,30,31,32)(27,35,29,36);(1,3,4,2)(5,9,10,7)(6,8,12,11)(13,15)(14,18)(16,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgh,d^2=f,e^2=fgh,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, 1032866);