Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1099 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)(5,6,10,15)(7,16,20,30)(8,12,22,29)(9,13,23,14)(11,18,25,19)(17,32,35,31)(21,27,36,28)(24,34,26,33);(5,7)(6,11)(8,17)(9,18)(10,20)(12,26)(13,16)(14,30)(15,25)(19,23)(21,33)(22,35)(24,29)(27,32)(28,31)(34,36);(5,8)(6,12)(7,17,20,35)(9,21)(10,22)(11,24,25,26)(13,27)(14,28)(15,29)(16,31,30,32)(18,33,19,34)(23,36);(6,13)(7,18)(9,23)(12,27)(14,15)(16,30)(17,33)(19,20)(21,36)(28,29)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291364);