Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,13,9,14)(10,15)(11,16)(12,19,26,20)(18,25)(21,31)(22,28,23,29)(24,30)(27,33,36,34)(32,35);(2,4)(5,7,10,21,11,18,24,32)(6,12,15,27,16,26,30,36)(8,20,22,34,9,19,23,33)(13,25,28,35,14,17,29,31);(6,13)(7,18)(8,9)(12,17)(14,16)(15,28)(21,32)(22,23)(25,26)(27,31)(29,30)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=ci,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72337456, #72337457, #72337464, … |
| maximal normal subgroup of | #72337456, #72337457, #72337464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53921);