Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,13,10,15,19,14)(7,12,16,11,17,20,8,18);(1,2,3,4);(6,11)(7,16)(8,17)(9,19)(12,13)(14,20)(15,18);(5,7,9,16,10,17,19,8)(6,12,13,11,15,20,14,18)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214781, #72216367, #72216494, … |
| maximal normal subgroup of | #72214781, #72216367, #72216494, … |
| maximal quotient of | #72214602, #72216334, #72216367, … |
| automorphism group of | #12068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403039);