Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2104 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)(5,6,9,8)(7,11,12,10)(13,14,18,19)(15,20,17,16)(21,22,25,29)(23,27,31,26)(24,28,33,36)(30,35,32,34);(6,8)(10,11)(14,19)(15,17)(21,23,25,31)(22,26,29,27)(24,32,33,30)(28,35,36,34);(7,12)(10,11)(13,15,18,17)(14,20,19,16)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,35)(33,36);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,16,18,20)(14,15,19,17)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,33)(34,36);(5,8,9,6)(7,11,12,10)(13,17,18,15)(14,20,19,16)(21,24,25,33)(22,28,29,36)(23,30,31,32)(26,34,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374261);