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 | |
1896 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)(5,6,10,8)(7,9,12,11)(13,14,16,18)(15,19,20,17)(21,22)(23,29)(24,26)(25,32)(27,31)(28,33)(30,35)(34,36);(5,6,10,8)(7,9,12,11)(15,20)(17,19)(22,24)(25,33)(27,29)(30,34);(5,7,10,12)(6,11,8,9)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,29)(25,34)(26,31)(28,36)(30,33)(32,35);(1,2,3,4)(5,8,10,6)(7,9,12,11)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,35,34,36);(5,9)(6,7)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=h,d^2=hi,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, 4009123);