Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#106 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,12,10,15,22,14)(7,16,19,33,20,34,17,31)(9,23,21,13,25,30,35,32)(11,26,28,18,29,24,27,36);(5,7,10,20)(6,11)(8,17,22,19)(9,24,25,26)(12,27)(13,31)(14,28)(15,29)(16,23)(18,35,36,21)(30,34)(32,33);(2,4);(5,8,10,22)(6,12,15,14)(7,17,20,19)(9,21,25,35)(11,27,29,28)(13,30,32,23)(16,31,34,33)(18,26,36,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237237, #72239532, #72239536, … |
| maximal normal subgroup of | #72237237, #72239532, #72239536, … |
| maximal quotient of | #72309900, C4×#12929, #72309940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401462);