Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1886 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)(5,6)(7,11)(8,9)(10,12)(13,14,19,20)(15,16,18,17)(22,26)(23,29)(24,31)(34,36);(5,7)(6,10)(8,11)(9,12)(13,15,19,18)(14,17,20,16)(22,26)(23,29)(28,33)(30,35);(6,8)(10,11)(13,16,19,17)(14,15,20,18)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,17,19,16)(14,15,20,18)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,8,9,6)(7,11,12,10)(13,18,19,15)(14,16,20,17)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=fgh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2216520);