Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135117 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #463 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,14)(9,18,25,21)(13,22,29,32)(15,23)(16,27)(26,28,31,30)(33,34)(35,45,49,40)(36,51,52,83)(37,55)(38,42)(39,60)(41,53,65,85)(43,59)(44,46)(47,75)(48,79,56,89)(50,64,73,72)(54,69,61,84)(57,70,87,66)(58,88)(62,77)(63,81,68,91)(67,74)(71,86)(76,92)(78,96)(80,93)(82,94)(90,95);(1,3,6,15,7,16,5,14)(2,8,20,27,11,23,10,12)(4,13,21,30,19,29,18,28)(9,22,17,31,25,32,24,26)(33,35,46,73,38,49,39,50)(34,40,44,64,42,45,60,72)(36,47,54,78,52,76,61,82)(37,56,74,81,58,48,59,91)(41,62,70,95,65,71,66,93)(43,68,55,79,67,63,88,89)(51,75,69,96,83,92,84,94)(53,86,87,80,85,77,57,90);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,28)(15,30)(16,29)(17,20)(23,32)(27,31)(33,36,38,52)(34,41,42,65)(35,47,49,76)(37,57,58,87)(39,61,46,54)(40,62,45,71)(43,69,67,84)(44,70,60,66)(48,80,56,90)(50,82,73,78)(51,55,83,88)(53,59,85,74)(63,94,68,96)(64,95,72,93)(75,79,92,89)(77,81,86,91)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,c^2=i,d^2=gi,e^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=egh,f^c=fg,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42317);