Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3415 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
979 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)(5,6,10,14,11,8,23,15)(7,16,20,18,22,34,21,35)(9,25,27,13,28,31,36,24)(12,29,30,33,17,26,32,19);(5,7)(6,12)(8,17)(9,26)(10,20)(11,22)(13,18)(14,30)(15,32)(16,25)(19,27)(21,23)(24,35)(28,29)(31,34)(33,36);(2,4)(5,8)(6,11)(7,17)(9,24)(10,14)(12,22)(13,28)(15,23)(16,33)(18,29)(19,34)(20,30)(21,32)(25,36)(26,35)(27,31);(5,6)(7,18)(8,11)(9,25)(10,14)(12,26)(13,27)(15,23)(16,21)(17,29)(19,30)(20,34)(22,35)(24,36)(28,31)(32,33)Polycyclic presentation:
2^9|a^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216402, #72225566, #72225568 |
| maximal normal subgroup of | #72216402, #72225566, #72225568 |
| maximal quotient of | #72226890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404655);