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)(35,47,52,72)(36,54)(37,59)(38,43)(39,62)(40,46)(41,50,65,80)(42,56)(44,60)(45,61)(48,70)(49,63,78,89)(51,66,74,88)(53,64,81,73)(55,68)(57,87)(58,84)(67,86)(69,75)(71,91,92,77)(76,93,94,82)(79,96,95,90)(83,85);(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)(33,35,48,74,61,81,60,80,40,52,38,51,39,53,37,50)(34,41,59,73,62,88,43,47,46,65,44,64,45,66,70,72)(36,49,75,91,86,95,85,94,58,78,56,77,57,79,55,76)(42,63,84,93,83,96,67,71,69,89,54,82,68,90,87,92);(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)(33,36,38,56,61,86,37,55,40,58,48,75,39,57,60,85)(34,42,59,84,62,83,43,67,46,69,44,54,45,68,70,87)(35,49,51,77,81,95,50,76,52,78,74,91,53,79,80,94)(41,63,73,93,88,96,47,71,65,89,64,82,66,90,72,92)Polycyclic presentation:
2^9|a^2=f,b^2=ei,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, 42890);