Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486827 |
| 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×C22⋊C4 | C24 |
| prime radical | #3459 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 747 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18)(19,27)(20,24)(21,34)(22,26)(23,29)(25,37)(28,44)(30,43)(31,47)(32,48)(33,42)(35,39)(36,41)(38,45)(40,46);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,22,30)(18,23,26,39)(20,31,33,28)(21,35,37,29)(24,40,42,38)(25,43,34,27)(32,45,36,46)(41,44,48,47);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,22,33)(18,24,26,42)(19,28,30,31)(21,36,37,32)(23,38,39,40)(25,41,34,48)(27,44,43,47)(29,46,35,45);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394754);