Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1846 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)(7,11)(9,12)(14,17)(15,19)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,35,34,36);(5,6,10,8)(7,12,11,9)(15,19)(18,20)(22,26)(23,30)(24,33)(35,36);(5,7)(6,9)(8,12)(10,11)(13,14,16,17)(15,18,19,20)(21,23,25,30)(22,27,26,29)(24,34,33,31)(28,36,32,35);(1,2)(3,4)(5,8,10,6)(7,12,11,9)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,36)(34,35);(5,9,10,12)(6,11,8,7)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,35)(29,36)(30,34)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3951666);