Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1737 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)(5,6,10,16)(7,18,21,31)(8,23,25,13)(9,15,26,14)(11,17,28,34)(12,29,32,19)(20,35,24,33)(22,36,27,30);(2,4)(5,7)(6,12,17,33)(8,24,26,19)(9,20)(10,27,28,21)(11,22)(13,30,15,31)(14,18)(16,32)(23,36)(25,29)(34,35);(6,13)(9,25)(10,28)(12,30)(14,16)(15,17)(18,32)(20,29)(21,27)(23,34)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,e^c=ei,f^c=fi,f^d=fi,g^d=gi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280569, #72280625, #72280636, … |
| maximal normal subgroup of | #72280569, #72280625, #72280636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53783);