Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C4⋊C4 |
| prime radical | C23 | #405 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,14,18,13,17,26,12,15,19,31,27,29,36,40,28)(11,20,23,21,24,16,32,33,25,34,30,39,22,35,38,37);(1,2,3,5,4,6,7,8)(9,11)(10,16)(12,21)(13,24)(14,30)(15,25)(17,34)(18,37)(19,35)(20,36)(22,29)(23,31)(26,38)(27,33)(28,39)(32,40);(9,12,26,36,29,18,14,19,15,28,40,17,13,27,31,10)(11,21,38,20,22,37,30,35,25,39,32,34,24,33,23,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^b=di,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72223649, #72223664, C8×#2171, … |
| maximal normal subgroup of | #72223649, #72223664, C8×#2171, … |
| maximal quotient of | C2×#129044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36240);