Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1835 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #488 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2)(3,11,14,7)(4,15,17,30)(5,10)(6,9)(8,16,21,31)(12,25)(13,24,28,20)(18,22)(19,26)(23,32)(27,29)(33,34)(35,47,51,72)(36,53)(37,44)(38,60)(39,62)(40,46)(41,49,65,79)(42,56)(43,54)(45,61)(48,73,77,63)(50,80,74,66)(52,82,81,64)(55,84)(57,67)(58,86)(59,70)(68,88)(69,87)(71,91,92,76)(75,93,95,83)(78,90,96,94)(85,89);(1,3,5,13,6,14,18,28)(2,7,10,20,9,11,22,24)(4,12,16,26,17,27,31,32)(8,19,15,29,21,23,30,25)(33,35,38,50,61,81,59,79,40,51,54,74,39,52,37,49)(34,41,44,64,62,80,43,47,46,65,70,82,45,66,60,72)(36,48,56,76,89,96,88,95,58,77,87,91,57,78,55,75)(42,63,53,83,84,94,67,71,69,73,86,93,68,90,85,92);(1,4,6,17)(2,8)(3,12,14,27)(5,16,18,31)(7,19)(9,21)(10,15)(11,23)(13,26,28,32)(20,29)(22,30)(24,25)(33,36,54,87,61,89,37,55,40,58,38,56,39,57,59,88)(34,42,44,53,62,84,43,67,46,69,70,86,45,68,60,85)(35,48,74,91,81,96,49,75,51,77,50,76,52,78,79,95)(41,63,64,83,80,94,47,71,65,73,82,93,66,90,72,92)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=dg,d^2=h,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fgh,f^c=fghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42889);