Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146102 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #490 |
| 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 | |
170 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,6,11,7,12,5,10)(3,13,16,8,17,24,15,25)(4,18,20,26,21,27,22,9)(14,29,30,32,19,23,31,28)(33,34)(35,45,50,40)(36,51,56,69)(37,43)(38,44)(39,42)(41,53,67,55)(46,77)(47,65,49,63)(48,76,82,64)(52,86)(54,85,89,74)(57,72)(58,70)(59,92,80,73)(60,68,88,95)(61,71)(62,78)(66,81)(75,83)(79,93)(84,87)(90,94)(91,96);(1,3,7,17)(2,8,12,25)(4,19,21,14)(5,16,6,15)(9,23,26,29)(10,24,11,13)(18,32,27,28)(20,30,22,31)(33,35)(34,40)(36,52)(37,47)(38,59)(39,50)(41,66)(42,45)(43,63)(44,73)(46,54)(48,57)(49,58)(51,84)(53,81)(55,90)(56,87)(60,83)(61,80)(62,68)(64,72)(65,70)(67,94)(69,86)(71,92)(74,77)(75,95)(76,96)(78,88)(79,89)(82,91)(85,93);(1,4,5,20,7,21,6,22)(2,9,10,18,12,26,11,27)(3,14,15,30,17,19,16,31)(8,23,13,28,25,29,24,32)(33,36)(34,41,42,67)(35,46)(37,53)(38,60)(39,56)(40,62,45,75)(43,51,70,69)(44,74,71,85)(47,78)(48,81)(49,83)(50,79)(52,59)(54,57)(55,58)(61,88)(63,93,65,77)(64,84,76,86)(66,92,94,73)(68,96,95,72)(80,87)(82,90)(89,91)Polycyclic presentation:
2^9|a^2=ehi,b^2=g,c^2=d,d^2=g,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=eg,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, 53305);