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×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2278 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)(6,9)(10,12)(13,14,17,19)(15,16,20,18)(21,22,26,27)(23,31,34,28)(24,29,30,25)(32,33,36,35);(7,11)(10,12)(13,14,17,19)(15,16,20,18)(22,27)(25,29)(28,31)(33,35);(5,6,8,9)(7,10,11,12)(13,15,17,20)(14,18,19,16)(21,23,26,34)(22,28,27,31)(24,32,30,36)(25,33,29,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,29)(23,32)(25,27)(26,30)(28,35)(31,33)(34,36);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,29)(22,30,27,24)(23,33,34,35)(28,36,31,32)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=g,e^2=f,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, 5516148);