Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,18,20,33)(8,22,21,13)(9,15)(10,25,27,16)(11,29)(12,17)(14,23)(19,34)(24,36,28,30)(26,32)(31,35);(1,2,3,4)(5,7,11,21,12,20,9,8)(6,13,15,18,17,22,29,33)(10,19,28,26,27,35,24,23)(14,30,31,25,32,36,34,16);(5,8,9,20,12,21,11,7)(6,14,15,31,17,32,29,34)(10,26,24,19,27,23,28,35)(13,30,18,25,22,36,33,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311922, #72311927, #72311949, … |
| maximal normal subgroup of | #72311922, #72311927, #72311949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54068);