Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | D4×C16 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,9,16,10,18,26,41,11,15,27,38,29,39,51,61,12,17,28,40,30,42,52,62,8,13,24,32,25,33,50,55)(7,14,21,35,22,37,47,59,23,34,48,56,49,57,65,67,19,36,43,58,44,60,63,68,20,31,45,53,46,54,64,66);(2,4)(6,13,17,15)(7,19)(14,34,36,31)(16,32,40,38)(18,33,42,39)(20,23)(21,43)(22,44)(35,56,58,53)(37,57,60,54)(41,55,62,61)(45,48)(46,49)(47,63)(59,67,68,66)(64,65);(5,7,9,21,10,22,26,47,11,23,27,48,29,49,51,65,12,19,28,43,30,44,52,63,8,20,24,45,25,46,50,64)(6,14,16,35,18,37,41,59,15,34,38,56,39,57,61,67,17,36,40,58,42,60,62,68,13,31,32,53,33,54,55,66)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=di,f^a=fi,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72287227 |
| maximal normal subgroup of | #72287227 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59097);