Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 52 points:
(2,4)(5,6,9,11,10,14,8,13)(7,15,17,12,18,19,16,20)(21,22)(23,31,33,48)(24,27)(25,37)(26,40)(28,32,42,50)(29,39)(30,36)(34,45,41,49)(35,47)(38,46,52,44)(43,51);(1,2,3,4)(6,11)(8,9)(12,15)(13,14)(16,17)(19,20)(22,27)(25,35)(28,42)(30,40)(32,50)(38,52)(39,51)(44,46);(5,7,10,18)(6,12,14,20)(8,17,9,16)(11,15,13,19)(21,23)(22,28)(24,33)(25,38)(26,41)(27,42)(29,45)(30,46)(31,47)(32,51)(34,36)(35,52)(37,48)(39,50)(40,44)(43,49)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=g,d^2=g,e^2=gh,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44722);