Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139074 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #459 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1452 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30)(33,34,39,45,40,46,62,78,41,47,38,63,61,77,37,44)(35,48,52,71,53,76,50,79,54,82,51,42,66,69,64,70)(36,55,57,80,58,91,56,43,59,73,60,74,67,72,65,75)(49,83,85,90,86,68,88,81,87,94,92,96,89,95,84,93);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,17,28,16,27)(8,21,24,25,15,31,23,32)(33,35,40,66,41,54,61,53)(34,42,46,79,47,71,77,70)(36,49,67,86,59,87,58,89)(37,51,39,50,62,52,38,64)(43,68,80,94,75,95,74,83)(44,76,45,48,78,69,63,82)(55,90,72,81,73,96,91,93)(56,85,57,88,65,92,60,84);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,29)(14,28)(16,20)(18,23)(22,31)(26,32)(27,30)(33,36,41,59)(34,43,47,75)(35,49,54,87)(37,60,62,57)(38,65,39,56)(40,67,61,58)(42,68,71,95)(44,55,78,73)(45,72,63,91)(46,80,77,74)(48,81,82,93)(50,85,64,92)(51,84,52,88)(53,89,66,86)(69,96,76,90)(70,83,79,94)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=ehi,f^c=fgh,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46271);