Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
2090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,8)(7,11)(14,18)(16,19)(21,22,26,29)(23,30,27,24)(25,33,35,28)(31,32,34,36);(2,4)(21,23,26,27)(22,24,29,30)(25,34,35,31)(28,36,33,32);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,24)(25,31)(26,27)(28,32)(29,30)(33,36)(34,35);(5,7)(6,10)(8,12)(9,11)(13,15,17,20)(14,19,18,16)(21,24)(22,27)(23,29)(25,32)(26,30)(28,34)(31,33)(35,36);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,20,18,15)(21,25,26,35)(22,28,29,33)(23,31,27,34)(24,32,30,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1126172);