Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10026 |
| 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 | |
1272 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,3)(2,8)(4,12)(5,9)(6,14)(7,15)(10,13)(11,16)(22,31)(25,32);(1,4)(2,9)(3,11)(5,16)(6,13)(7,8)(10,15)(12,14)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(1,5,7,6)(2,10,12,11)(3,9,15,14)(4,16,8,13)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139872);