Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| 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 | C22×C4×D4 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(3,10)(7,14)(11,16)(13,15)(19,25)(22,28)(26,31)(29,32);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dhi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72307005, #72307008, #72307011, … |
| maximal normal subgroup of | #72307005, #72307008, #72307011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252142);