Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| 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 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,9,12,10,7,8,11)(13,14,18,24,19,25,16,22)(15,26,29,20,30,38,27,39)(17,32,34,40,35,21,31,23)(28,41,43,36,33,37,42,44);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,29)(17,33)(18,27)(19,30)(21,36)(22,38)(23,41)(24,26)(25,39)(28,35)(31,42)(32,44)(34,43)(37,40);(5,6,10,7)(8,12,9,11)(14,21)(16,18)(17,34)(20,36)(22,23)(24,40)(25,32)(26,37)(27,29)(28,42)(31,35)(33,43)(38,41)(39,44)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=gi,d^2=g,e^2=ghi,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44764);