Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | Aut(G) order 98304 |
| 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 | #62810 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,12,9,13,15,16)(17,18,22,24,23,19,20,26)(21,25,30,31,29,27,28,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32);(2,7)(3,9)(8,14)(10,15)(17,20,23,22)(18,24,19,26)(21,28,29,30)(25,31,27,32);(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32);(21,29)(25,27)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,c^b=ci,f^b=fh,e^d=eh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250181);