Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1822 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)(5,6,9,11)(7,12,8,10)(13,14,18,16)(15,19,17,20)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35);(5,7,9,8)(6,10,11,12)(13,15,18,17)(14,19,16,20)(23,30)(25,33)(27,29)(28,32);(7,8)(10,12)(13,16,18,14)(15,19,17,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,17,18,15)(14,19,16,20)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(5,8,9,7)(6,10,11,12)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4052501);