Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487707 |
| 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×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,9)(5,11)(6,10)(7,12)(8,14)(15,16)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(19,26)(20,28)(23,30)(29,31);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(18,22)(19,26)(20,28)(23,31)(24,25)(29,30);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(17,19,21,26)(18,23,25,30)(20,27,28,32)(22,29,24,31)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,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=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395627);