Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2024 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)(5,6,9,12)(7,10,11,8)(21,22)(23,28)(24,33)(25,30)(26,27)(29,34)(31,35)(32,36);(5,7,9,11)(6,10,12,8)(13,14,16,18)(15,17,19,20)(22,27)(23,31)(29,33)(30,32);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,31)(22,28,27,35)(24,32,34,30)(25,33,36,29);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(21,24,26,34)(22,29,27,33)(23,32,31,30)(25,35,36,28);(5,8,9,10)(6,11,12,7)(13,15,16,19)(14,17,18,20)(21,25)(22,30)(23,33)(24,28)(26,36)(27,32)(29,31)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1205139);