Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486155 |
| 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×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1006 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)(3,13)(4,9)(5,10)(6,14)(7,23)(8,15)(11,16)(12,19)(17,29)(18,28)(20,30)(21,27)(22,26)(24,32)(25,31);(2,9)(3,16)(4,19)(5,18)(10,24)(11,29)(12,14)(13,25)(15,27)(20,21)(23,30)(26,28);(2,10)(3,5)(4,20)(9,24)(11,23)(12,26)(13,15)(14,28)(16,18)(17,31)(19,21)(22,32)(25,27)(29,30);(1,3,8,5)(2,11,15,28)(4,17,21,22)(6,16,7,18)(9,25,27,24)(10,14,13,23)(12,29,30,26)(19,31,20,32)Polycyclic presentation:
2^9|d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251767 |
| maximal normal subgroup of | #72251767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393352);