Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1919 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
540 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)(5,6,10,16,12,17,11,26)(7,18,21,34,20,35,27,29)(8,14,22,33,23,15,9,24)(13,28,31,36,30,25,32,19);(5,7)(6,13,17,30)(8,19)(9,25)(10,21)(11,27)(12,20)(14,29,15,34)(16,31,26,32)(18,24,35,33)(22,28)(23,36);(2,4)(5,8,10,22,12,23,11,9)(6,14,16,33,17,15,26,24)(7,19,21,28,20,36,27,25)(13,29,31,18,30,34,32,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,e^b=ei,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54422);