Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143663 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #493 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1452 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,7,12)(3,13,16,24)(4,9,20,27)(5,11,6,10)(8,17,23,15)(14,29,30,25)(18,32,31,22)(19,28,21,26)(33,34,40,47)(35,48,53,82)(36,54,59,91)(37,44,62,79)(38,43,63,80)(39,45,61,46)(41,66,71,50)(42,72,77,57)(49,73,85,95)(51,70,60,69)(52,68,64,78)(55,92,88,67)(56,76,93,75)(58,74,65,90)(81,86,89,84)(83,94,87,96);(1,3)(2,8)(4,18)(5,17)(6,15)(7,16)(9,25)(10,24)(11,13)(12,23)(14,19)(20,31)(21,30)(22,26)(27,29)(28,32)(33,35,40,53)(34,41,47,71)(36,55,59,88)(37,60,62,51)(38,50,63,66)(39,64,61,52)(42,73,77,95)(43,78,80,68)(44,48,79,82)(45,70,46,69)(49,65,85,58)(54,89,91,81)(56,87,93,83)(57,86,72,84)(67,76,92,75)(74,94,90,96);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,28)(12,27)(13,25)(15,18)(16,30)(17,31)(23,32)(24,29)(33,36,40,59)(34,42,47,77)(35,49,53,85)(37,57,62,72)(38,56,63,93)(39,65,61,58)(41,67,71,92)(43,54,80,91)(44,74,79,90)(45,76,46,75)(48,81,82,89)(50,86,66,84)(51,87,60,83)(52,88,64,55)(68,96,78,94)(69,95,70,73)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fgh,g^c=gi,h^c=hi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50879);