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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2125 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)(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,17,19,20)(23,27)(24,31)(30,35)(32,36);(5,6)(7,9)(8,10)(11,12)(15,19)(17,20)(21,22,26,29)(23,24,27,31)(25,33,34,28)(30,36,35,32);(7,12)(9,11)(13,15,16,19)(14,17,18,20)(21,23)(22,24)(25,30)(26,27)(28,32)(29,31)(33,36)(34,35);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(21,24)(22,27)(23,29)(25,32)(26,31)(28,35)(30,33)(34,36);(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,34)(22,28,29,33)(23,30,27,35)(24,32,31,36)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=h,e^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, 5518628);