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 | |
2044 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)(7,9)(8,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,27,30)(23,24,25,28)(26,34,36,29)(31,35,33,32);(8,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,23,27,25)(22,28,30,24)(26,33,36,31)(29,32,34,35);(6,7)(10,11)(13,14,16,18)(15,17,20,19)(21,24,27,28)(22,23,30,25)(26,35,36,32)(29,33,34,31);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,25,27,23)(22,24,30,28)(26,31,36,33)(29,35,34,32);(5,8)(6,10)(7,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,36)(28,35)(30,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fgh,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, 4049476);