Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343524 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62818 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7,9,20)(4,15)(8,19,24,31)(10,21,11,16)(14,27)(17,29)(22,28,23,30)(26,32);(1,2)(3,8)(4,16)(5,10)(6,11)(7,17)(9,18)(12,22)(13,23)(14,28)(15,20)(19,26)(21,29)(24,25)(27,31)(30,32);(1,3)(2,8)(4,14)(5,13)(6,12)(7,19)(9,24)(10,23)(11,22)(15,27)(16,28)(17,26)(18,25)(20,31)(21,30)(29,32);(3,12)(8,22)(13,25)(14,26)(19,28)(23,24)(27,32)(30,31);(3,13)(8,23)(12,25)(14,27)(19,30)(22,24)(26,32)(28,31)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250720);