Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,23)(18,32)(19,28)(21,27)(24,30)(25,29)(26,33)(31,34)(35,36);(5,7)(6,12,16,27)(8,18)(9,20)(10,21)(11,19)(13,26,29,32)(14,17,15,28)(22,31,30,36)(23,34)(24,33)(25,35);(1,3,4,2)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,34)(21,35)(27,32)(28,36);(7,19)(12,27)(17,28)(18,33)(20,21)(26,32)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72316894, #72316899, #72316909, … |
| maximal normal subgroup of | #72316894, #72316899, #72316909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261112);