Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,8,22,15)(7,16,20,32,17,35,21,18)(9,25,28,13,29,27,33,24)(12,30,31,36,23,26,34,19);(5,7)(6,12)(8,23)(9,26)(10,20)(11,17)(13,32)(14,31)(15,34)(16,25)(18,24)(19,28)(21,22)(27,35)(29,30)(33,36);(2,4)(7,17)(12,23)(16,35)(18,32)(19,36)(20,21)(26,30)(31,34);(5,8)(6,11)(7,18)(9,27)(10,15)(12,30)(13,33)(14,22)(16,20)(17,32)(19,34)(21,35)(23,26)(24,28)(25,29)(31,36)Polycyclic presentation:
2^9|a^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404645);