Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196092 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,20,22,30)(9,24)(10,13)(11,15)(12,18)(14,23,29,21)(17,32,34,27)(19,28)(25,26)(31,33,36,35);(5,7,9,18,10,19,11,25)(6,12,15,16,13,26,24,28)(8,17,23,35,22,34,21,33)(14,27,30,31,29,32,20,36);(1,2,3,4)(6,13)(12,26)(14,29)(15,24)(16,28)(20,30)(27,32)(31,36);(5,8)(6,14)(7,17)(9,23)(10,22)(11,21)(12,27)(13,29)(15,30)(16,31)(18,35)(19,34)(20,24)(25,33)(26,32)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300257);