Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,7)(8,12)(14,18)(15,20)(21,22,26,27)(23,24,30,28)(25,29,34,35)(31,32,33,36);(5,6,9,7)(8,11,12,10)(13,14,17,18)(15,19,20,16)(22,27)(23,30)(29,35)(31,33);(8,12)(10,11)(14,18)(16,19)(21,23,26,30)(22,28,27,24)(25,33,34,31)(29,32,35,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,19,18,16)(21,24,26,28)(22,23,27,30)(25,32,34,36)(29,31,35,33);(5,8,9,12)(6,10,7,11)(13,16,17,19)(14,20,18,15)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5381351);