Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fhi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144571);