Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2050 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)(7,11,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,25)(26,27)(28,31)(29,34)(32,33)(35,36);(7,8)(10,11)(13,15,16,19)(14,17,18,20)(22,27)(23,31)(24,34)(33,35);(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,28)(24,32)(25,35)(26,31)(27,30)(29,33)(34,36);(1,2,3,4)(5,8,9,7)(6,11,12,10)(21,24)(22,29)(23,32)(25,27)(26,34)(28,33)(30,35)(31,36);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,29)(22,24,27,34)(23,33,31,35)(28,36,30,32)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4572549);