Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486084 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18186 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
658 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)(3,15)(4,19)(5,11)(6,24)(7,26)(8,14)(9,17)(10,21)(12,25)(13,22)(16,31)(18,30)(20,28)(23,29)(27,32);(1,3)(2,9)(4,16)(5,22)(6,21)(7,23)(8,18)(10,13)(11,28)(12,29)(14,27)(15,24)(17,25)(19,26)(20,32)(30,31);(1,4,8,21)(2,10)(3,16,18,6)(5,20,23,25)(7,17,22,32)(9,13)(11,12)(14,19)(15,24)(26,27)(28,29)(30,31);(1,5,22,3,8,23,7,18)(2,11,13,15,14,29,26,30)(4,20,32,16,21,25,17,6)(9,24,19,28,27,31,10,12)Polycyclic presentation:
2^9|c^2=h,d^2=gh,e^2=gi,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72231710, #72250033, C2×#486084, … |
| maximal normal subgroup of | #72231710, #72250033, C2×#486084, … |
| maximal quotient of | #72250333, #72250343, C2×#486084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393280);