Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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×C22⋊C4 |
| prime radical | #62800 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3069 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)(3,8)(4,6)(5,7)(9,12)(10,14)(11,13)(15,16)(17,18,20,24,21,23,27,30)(19,22,26,28,25,29,31,32);(2,6)(4,12)(7,9)(8,13)(11,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(2,7)(6,9)(8,14)(13,15)(18,23)(19,25)(24,30)(26,31);(1,3)(2,8)(4,11)(5,10)(6,13)(7,14)(9,15)(12,16);(3,10)(8,14)(11,16)(13,15)Polycyclic presentation:
2^9|a^2=g,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,c^b=ci,f^b=fh,e^d=eh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250182);