Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,14)(8,12)(9,11)(10,15)(13,17)(16,19)(18,20)(21,22,26,29,25,24,31,28)(23,30,33,35,32,27,34,36);(2,3)(5,7,8,15)(6,10,12,14)(9,16,17,20)(11,18,13,19)(21,23,31,34,25,32,26,33)(22,27,28,35,24,30,29,36);(6,11)(9,17)(10,18)(12,13)(14,19)(16,20)(21,24)(22,25)(23,30)(26,28)(27,32)(29,31)(33,35)(34,36);(1,3,4,2)(7,15)(10,14)(16,20)(18,19)(22,24)(23,32)(28,29)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,d^c=di,f^c=fh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136947);