Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62686 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2976 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)(17,18,22,26,23,19,21,25)(20,27,30,31,28,24,29,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,24)(21,26)(22,25)(27,28)(29,31)(30,32);(2,7)(3,9)(8,14)(10,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32);(20,28)(24,27)(29,30)(31,32);(17,21,23,22)(18,25,19,26)(20,29,28,30)(24,31,27,32)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,c^b=ci,f^b=fh,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250175);