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 | |
2056 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)(8,10)(11,12)(15,20)(16,18)(21,22,26,27)(23,24,29,28)(25,30,33,35)(31,32,36,34);(5,7,8,11)(6,9,10,12)(13,14,17,19)(15,16,20,18)(22,27)(23,29)(25,33)(32,34);(6,10)(9,12)(13,15,17,20)(14,18,19,16)(21,23,26,29)(22,28,27,24)(25,31,33,36)(30,34,35,32);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,26,28)(22,29,27,23)(25,34,33,32)(30,31,35,36);(5,7)(6,9)(8,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,35)(28,34)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,d^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, 5517102);