Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,21)(5,13)(6,24)(7,26)(8,15)(9,27)(10,16)(11,18)(12,17)(19,28)(20,31)(22,29)(23,30)(25,32);(1,3,9,20)(2,11,27,21)(4,17,18,25)(5,23,22,6)(7,8,19,10)(12,28,32,26)(13,16,29,15)(14,30,31,24);(1,4)(2,12)(3,17)(5,10)(6,19)(7,23)(8,22)(9,18)(11,28)(13,14)(15,24)(16,30)(20,25)(21,26)(27,32)(29,31);(1,5,8,18)(2,13,15,11)(3,17,19,6)(4,9,22,10)(7,23,20,25)(12,28,24,14)(16,21,27,29)(26,30,31,32)Polycyclic presentation:
2^9|b^2=gi,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394643);