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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2266 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)(21,22,25,26)(23,27,29,33)(24,31,32,28)(30,36,35,34);(6,9)(10,12)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,31)(30,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,31)(25,26)(28,32)(29,33)(30,36)(34,35);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,17,18,20)(21,23,25,29)(22,27,26,33)(24,30,32,35)(28,34,31,36);(21,24,25,32)(22,28,26,31)(23,30,29,35)(27,34,33,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=f,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, 674279);