Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143516 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19258 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 48 points:
(1,2,5,9)(3,10,13,15)(4,7,14,8)(6,11,16,12)(17,18)(19,27)(20,31)(21,36)(22,25)(23,28)(24,34)(26,33)(29,43)(30,45)(32,42)(35,46)(37,40)(38,44)(39,47)(41,48);(1,3)(2,6)(4,11)(5,13)(7,10)(8,15)(9,16)(12,14)(17,19)(18,23)(20,32)(21,37)(22,29)(24,40)(25,38)(26,42)(27,31)(28,34)(30,33)(35,47)(36,39)(41,45)(43,46)(44,48);(2,7)(3,11)(4,14)(8,9)(10,15)(12,13)(23,38)(27,43)(39,42)(40,45)Polycyclic presentation:
2^9|a^2=h,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fh47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50760);