Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143528 |
| 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 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 374 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18)(19,30)(20,24)(21,29)(22,27)(23,28)(25,32)(26,34)(31,45)(33,44)(35,40)(36,39)(37,43)(38,46)(41,48)(42,47);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(18,24)(20,34)(21,22)(25,41)(26,29)(27,40)(28,39)(30,45)(31,46)(32,37)(35,36)(38,43)(42,44)(47,48);(2,8)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19)(18,25)(20,31)(21,37)(22,32)(23,33)(24,38)(26,42)(27,44)(28,43)(29,30)(34,46)(35,48)(36,47)(39,41)(40,45)Polycyclic presentation:
2^9|d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,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=ehi,f^c=fh41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50724);