Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| 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)(8,21)(9,14)(10,15)(11,16)(12,19)(13,22)(18,26)(20,27)(23,30)(24,29)(25,28)(31,33)(32,35)(34,36);(5,7)(6,12,16,28)(8,18)(9,19)(10,25)(11,20)(13,26,30,36)(14,27,15,17)(21,31,29,35)(22,32)(23,33)(24,34);(1,3,4,2)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,26)(14,29)(15,21)(16,30)(17,31)(19,32)(20,34)(25,33)(27,35)(28,36);(5,9,11,10)(6,14,16,15)(7,19,20,25)(8,22,24,23)(12,27,28,17)(13,29,30,21)(18,32,34,33)(26,35,36,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,f^b=fi,e^c=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72313078, #72313126, #72313176, … |
| maximal normal subgroup of | #72313078, #72313126, #72313176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261132);