Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,14)(10,22)(11,15)(12,19)(17,32)(18,27)(20,29)(21,31)(23,26)(24,30)(25,34)(28,35)(33,36);(5,7)(6,12)(8,17)(9,19)(10,23)(11,18)(13,28)(14,27)(15,26)(16,22)(20,34)(21,33)(24,35)(25,31)(29,32)(30,36);(2,4)(5,8)(6,13)(7,17)(9,20)(10,24)(11,21)(12,25)(14,29)(15,31)(16,32)(18,33)(19,34)(22,30)(23,35)(26,28)(27,36);(7,18)(12,26)(16,27)(17,33)(19,23)(25,28)(32,36)(34,35);(5,9,11,10)(6,14,15,22)(7,19,18,23)(8,20,21,24)(12,27,26,16)(13,29,31,30)(17,34,33,35)(25,36,28,32)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=di,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72234838, #72236660, #72238124, … |
| maximal normal subgroup of | #72234838, #72236660, #72238124, … |
| maximal quotient of | #72234697, #72240236, #72257308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338918);