Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
890 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)(7,16)(8,13)(9,12)(10,15)(11,18)(14,24)(17,31)(19,32)(20,33)(21,28)(22,29)(23,30)(25,36)(26,34)(27,35);(5,7,8,18)(6,11,13,16)(9,19,21,35)(10,20,23,36)(12,27,28,32)(14,26,29,31)(15,25,30,33)(17,22,34,24);(6,12)(7,17)(9,24)(11,25)(13,28)(14,15)(18,35)(19,20)(21,22)(29,30)(31,32)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311842, #72311880, #72311889, … |
| maximal normal subgroup of | #72311842, #72311880, #72311889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55451);