Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,22,26,23)(24,31,34,36)(25,30,32,28)(27,35,29,33);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,29,34,27)(25,30,32,28)(31,33,36,35);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,24,32,35,26,34,25,33)(22,27,30,36,23,29,28,31);(2,4)(7,11)(10,12)(13,15,16,19)(14,18,17,20)(22,23)(27,29)(28,30)(31,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376870);