Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #482 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
187 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,5,10,6,11,7,21)(3,12,14,24,16,28,15,8)(4,17,19,27,20,9,22,26)(13,29,18,32,30,25,31,23)(33,34,37,44,40,47,64,78,41,48,65,79,39,46,38,45)(35,49,51,71,55,69,53,72,56,82,89,70,54,68,52,42)(36,57,59,77,62,80,60,75,63,92,66,43,61,74,94,76)(50,83,84,73,87,67,85,95,88,96,90,81,86,91,58,93);(1,3,7,14,6,16,5,15)(2,8,21,12,11,24,10,28)(4,18,22,30,20,31,19,13)(9,25,27,23,17,29,26,32)(33,35)(34,42)(36,58,63,85)(37,52)(38,51)(39,55)(40,54)(41,56)(43,73,77,81)(44,68)(45,49)(46,71)(47,70)(48,72)(50,60,88,94)(53,65)(57,91,92,67)(59,86,66,87)(61,84,62,90)(64,89)(69,79)(74,83,80,96)(75,95,76,93)(78,82);(1,4)(2,9)(3,13)(5,19)(6,20)(7,22)(8,23)(10,26)(11,17)(12,29)(14,18)(15,31)(16,30)(21,27)(24,32)(25,28)(33,36,41,63)(34,43,48,77)(35,50,56,88)(37,60,65,94)(38,66,64,59)(39,62,40,61)(42,67,72,91)(44,57,79,92)(45,80,78,74)(46,76,47,75)(49,81,82,73)(51,85,89,58)(52,90,53,84)(54,87,55,86)(68,93,69,95)(70,96,71,83)Polycyclic presentation:
2^9|a^2=d,b^2=de,c^2=i,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fgi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53397);