Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2048 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)(6,9)(7,11)(14,17)(15,19)(21,22,27,30)(23,24,25,28)(26,29,34,36)(31,32,33,35);(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,17)(18,19)(21,23,27,25)(22,28,30,24)(26,31,34,33)(29,35,36,32);(6,9)(10,12)(15,19)(18,20)(21,24,27,28)(22,23,30,25)(26,32,34,35)(29,31,36,33);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,19)(17,20)(21,25,27,23)(22,24,30,28)(26,31,34,33)(29,35,36,32);(5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,17)(18,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2672211);