Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,4)(3,10,13,16)(5,9,6,15)(8,12,11,14)(17,18,21,26,22,25,20,24)(19,27,30,32,29,31,28,23);(1,3,6,14,7,13,5,12)(2,8,9,10,4,11,15,16)(17,19,20,28,22,29,21,30)(18,23,26,27,25,32,24,31);(1,4,7,2)(3,11,13,8)(5,9,6,15)(10,14,16,12)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,32,27);(1,5,7,6)(2,9,4,15)(3,12,13,14)(8,10,11,16)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,32,27);(3,13)(8,11)(10,16)(12,14)(18,25)(19,29)(24,26)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,e^b=eh,d^c=dhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72339789, #72339791, #72339800, … |
| maximal normal subgroup of | #72339789, #72339791, #72339800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251615);