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 | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,8,10,9,12,7,11)(13,14,16,21,31,40,17,33,19,24,32,41,18,23,30,22)(15,25,26,42,35,37,44,39,29,43,34,38,28,36,27,20);(1,2,3,4)(6,10)(7,8)(11,12)(13,15)(14,20)(16,27)(17,34)(18,35)(19,29)(21,36)(22,25)(23,42)(24,39)(26,30)(28,31)(32,44)(33,43)(37,41)(38,40);(14,21)(16,30)(17,32)(18,31)(20,36)(22,40)(23,33)(24,41)(25,38)(26,27)(28,35)(34,44)(37,39)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,e^2=h,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44145);