Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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×#2703 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2405 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)(13,14,17,16)(15,18,20,19)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35);(5,6,9,7)(8,11,12,10)(13,15,17,20)(14,18,16,19)(22,27)(23,29)(28,35)(30,36);(6,7)(10,11)(14,16)(18,19)(21,23,26,29)(22,24,27,32)(25,30,34,36)(28,31,35,33);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,17,14)(15,19,20,18)(21,24)(22,23)(25,33)(26,32)(27,29)(28,36)(30,35)(31,34);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,28)(23,30)(24,31)(26,34)(27,35)(29,36)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5513287);