Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,21,25,13)(9,26,28,36)(10,29)(11,16)(12,19)(14,30,31,27)(15,22)(18,35,24,32)(20,34)(23,33);(2,4)(5,7)(6,12)(8,9)(10,23)(11,20)(13,31)(14,21)(15,34)(16,33)(17,29)(18,27)(19,22)(24,30)(25,28)(26,32)(35,36);(1,2,3,4)(5,8,22,24,11,25,10,18)(6,13,29,32,16,21,15,35)(7,9,23,27,20,28,19,30)(12,31,34,36,33,14,17,26);(5,9)(6,14)(7,18)(8,23)(10,30)(11,28)(12,32)(13,17)(15,36)(16,31)(19,25)(20,24)(21,34)(22,27)(26,29)(33,35)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293128);