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)(35,47,51,41)(36,52,56,76)(37,44)(38,45)(39,46)(40,43)(42,54,68,63)(48,79)(49,75,62,65)(50,78,84,66)(53,86)(55,73,89,96)(57,71)(58,90,91,72)(59,92,93,69)(60,74)(61,70)(64,80)(67,87)(77,81)(82,94)(83,95)(85,88);(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)(34,41)(36,53)(37,49)(38,58)(39,62)(40,51)(42,67)(43,47)(44,65)(45,72)(46,75)(48,55)(50,60)(52,85)(54,87)(56,88)(57,84)(59,80)(61,91)(63,83)(64,69)(66,74)(68,95)(70,90)(71,78)(73,94)(76,86)(77,92)(79,96)(81,93)(82,89);(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)(34,42,43,68)(35,48)(37,54)(38,59)(39,63)(40,56)(41,64,47,77)(44,52,46,76)(45,73,70,96)(49,80)(50,83)(51,82)(53,58)(55,60)(57,89)(61,93)(62,81)(65,94,75,79)(66,86,78,85)(67,90,95,72)(69,71,92,74)(84,87)(88,91)Polycyclic presentation:
2^9|a^2=eh,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, 53304);