Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13960 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,10)(3,13,16,23,14,25,20,8)(4,17,18,24,19,26,21,9)(15,29,31,27,28,22,30,32);(1,3,7,20,6,14,5,16)(2,8,10,13,12,23,11,25)(4,15,21,31,19,28,18,30)(9,22,17,27,24,29,26,32);(3,14)(15,28)(16,20)(30,31);(1,4,7,21,6,19,5,18)(2,9,10,17,12,24,11,26)(3,15,20,31,14,28,16,30)(8,22,13,27,23,29,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285753, #72285922, #72286225, … |
| maximal normal subgroup of | #72285753, #72285922, #72286225, … |
| maximal quotient of | C2×#364127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271323);