Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C23 | #442 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,4,7,5,3,6,8)(9,10,15,21,16,22,38,54,17,23,14,39,25,44,13,20)(11,24,29,48,30,53,27,55,31,58,28,18,42,46,40,47)(12,32,34,56,35,67,33,19,36,50,37,51,43,49,41,52)(26,59,61,66,62,45,64,57,63,70,68,72,65,71,60,69);(1,3,4,8,5,2,6,7)(9,11,25,30,17,31,16,42)(10,18,44,47,23,48,22,55)(12,26,35,65,36,63,43,62)(13,28,14,40,38,29,15,27)(19,45,51,59,52,71,56,70)(20,53,39,58,54,46,21,24)(32,66,67,69,50,72,49,57)(33,61,37,60,41,68,34,64);(9,12,17,36)(10,19,23,52)(11,26,31,63)(13,37,38,34)(14,41,15,33)(16,43,25,35)(18,45,48,71)(20,32,54,50)(21,49,39,67)(22,56,44,51)(24,57,58,69)(27,61,40,68)(28,60,29,64)(30,65,42,62)(46,72,53,66)(47,59,55,70)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43888);