Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | |
1232 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)(8,11)(9,22)(10,21)(12,17)(13,27)(14,28)(16,29)(18,32)(19,26)(20,24)(23,33)(25,31)(30,35)(34,36);(5,7)(6,12)(8,19)(9,23)(10,26)(11,18)(13,30)(14,32)(15,21)(16,24)(17,28)(20,33)(22,36)(25,35)(27,34)(29,31);(2,4)(5,8,11,6)(7,12)(9,20,25,13)(10,21,28,14)(15,17)(16,29)(18,19)(22,27,31,24)(23,30)(26,32)(33,35)(34,36);(2,4)(5,9,11,25)(6,13,8,20)(7,16,18,34)(10,27,28,24)(12,29,19,36)(14,22,21,31)(15,33,32,30)(17,35,26,23)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,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, 296219);