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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2387 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)(5,6,9,7)(8,11,12,10)(14,17)(15,19)(21,22)(23,30)(24,25)(26,27)(28,33)(29,35)(31,36)(32,34);(5,6,9,7)(8,11,12,10)(15,19)(18,20)(22,27)(25,35)(28,30)(32,36);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,34)(25,36)(26,33)(27,30)(29,31)(32,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,25)(23,31)(26,29)(27,35)(28,32)(30,36)(33,34);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,29)(23,32)(24,27)(26,35)(28,34)(30,31)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fgh,d^2=h,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, 4052413);