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) = C26×C4 |
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 | |
2152 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)(6,7)(10,11)(13,14,17,20)(15,16,19,18)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35);(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,20)(17,19)(23,30)(25,33)(27,29)(28,32);(8,12)(10,11)(13,16,17,18)(14,19,20,15)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,16)(17,19)(18,20)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,20)(17,18)(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=fgh,c^2=gh,d^2=fh,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, 4052483);