Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2140 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)(13,14)(15,19)(16,18)(17,20)(21,22,27,30)(23,29,24,28)(25,26,35,36)(31,34,33,32);(8,12)(10,11)(13,15)(14,19)(16,17)(18,20)(21,23,27,24)(22,28,30,29)(25,31,35,33)(26,32,36,34);(6,7)(10,11)(14,16)(15,20)(21,24,27,23)(22,29,30,28)(25,31,35,33)(26,32,36,34);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,17,20,19)(21,25)(22,26)(23,31)(24,33)(27,35)(28,32)(29,34)(30,36);(5,8,9,12)(6,10,7,11)(13,17)(14,15)(16,20)(18,19)(21,26)(22,25)(23,32)(24,34)(27,36)(28,31)(29,33)(30,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=fg,e^2=fgh,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, 5385978);