Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1009 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)(7,18,21,33)(8,14,23,31)(9,15)(10,16)(11,17)(12,19)(13,22,30,25)(20,32,24,26)(27,34)(28,35)(29,36);(5,7,10,20,11,21,9,24)(6,12,16,28,17,29,15,27)(8,19,22,35,23,36,25,34)(13,26,14,18,30,32,31,33);(2,4)(5,8)(6,13)(7,19)(9,25)(10,22)(11,23)(12,26)(14,16)(15,31)(17,30)(18,28)(20,35)(21,36)(24,34)(27,33)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312034, #72312042, #72312062, … |
| maximal normal subgroup of | #72312034, #72312042, #72312062, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53833);