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×#388 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,19)(9,22)(10,14)(11,18)(12,21)(13,23)(16,29)(17,28)(20,26)(24,36)(25,32)(27,30)(31,33)(34,35);(5,7)(6,11,14,25)(8,16)(9,18)(10,17)(12,24,19,33)(13,15,22,28)(20,31)(21,30)(23,32)(26,27,34,29)(35,36);(5,8,9,20,10,21,23,35)(6,12,13,26,14,19,22,34)(7,16,18,31,17,30,32,36)(11,24,15,27,25,33,28,29);(7,17)(11,25)(15,28)(16,30)(18,32)(24,33)(27,29)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250753, #72250782, #72250791, … |
| maximal normal subgroup of | #72250753, #72250782, #72250791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260742);