Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
524 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)(7,17,20,33)(8,14)(9,25)(10,27)(11,16)(12,19,29,34)(13,24)(15,26)(18,30,22,36)(21,28,35,31)(23,32);(5,7,9,21,26,18,23,34,11,20,24,35,10,22,8,19)(6,12,14,30,27,28,13,17,16,29,32,36,15,31,25,33);(2,4)(7,18,20,22)(8,23)(9,24)(12,28,29,31)(13,25)(14,32)(17,30,33,36)(19,35,34,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249884, #72249887, #72249906, … |
| maximal normal subgroup of | #72249884, #72249887, #72249906, … |
| maximal quotient of | #72265682, C2×#147199 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54395);