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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(13,14,16,17)(15,19,20,18)(21,22)(23,28)(24,32)(25,35)(26,36)(27,29)(30,31)(33,34);(6,9)(10,12)(14,17)(18,19)(21,23,27,24)(22,28,29,32)(25,33,36,30)(26,31,35,34);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,17)(18,20)(21,24,27,23)(22,28,29,32)(25,33,36,30)(26,34,35,31);(1,2,3,4)(21,25)(22,26)(23,30)(24,33)(27,36)(28,34)(29,35)(31,32);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,26)(22,25)(23,31)(24,34)(27,35)(28,33)(29,36)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=gh,d^2=fgi,e^2=h,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, 2225444);