Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2060 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)(15,19)(17,20)(21,22,27,30)(23,25,24,28)(26,34,35,29)(31,36,32,33);(6,9)(10,12)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,24,27,23)(22,28,30,25)(26,31,35,32)(29,33,34,36);(13,15,16,19)(14,17,18,20)(21,25,27,28)(22,23,30,24)(26,33,35,36)(29,31,34,32);(5,7,8,11)(6,10,9,12)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 678813);