Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
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)(3,11,15,7)(4,16,18,31)(5,10)(6,9)(8,17,22,26)(12,19)(13,27)(14,24,25,21)(20,28)(23,32)(29,30)(33,34)(35,48,53,74)(36,56)(37,62)(38,64)(39,46)(40,45)(41,47)(42,51,66,52)(43,58)(44,63)(49,72)(50,79,82,91)(54,76,55,75)(57,71)(59,87)(60,86)(61,88)(65,84,90,83)(67,77,68,85)(69,89)(70,78)(73,80,92,81)(93,95,94,96);(1,3,12,25,6,15,5,14)(2,7,19,24,9,11,10,21)(4,13,26,32,18,29,17,28)(8,20,31,27,22,23,16,30)(33,35,49,77,40,55,38,52,41,53,63,85,39,54,37,51)(34,42,62,75,46,68,44,48,47,66,64,76,45,67,72,74)(36,50,78,95,60,84,58,81,61,82,89,96,59,83,57,80)(43,65,86,93,70,91,56,73,71,90,87,94,69,79,88,92);(1,4)(2,8,9,22)(3,13)(5,17)(6,18)(7,20,11,23)(10,16,19,31)(12,26)(14,28)(15,29)(21,30,24,27)(25,32)(33,36,37,57,39,59,63,89,41,61,38,58,40,60,49,78)(34,43,44,56,45,69,62,86,47,71,72,88,46,70,64,87)(35,50,51,80,54,83,85,96,53,82,52,81,55,84,77,95)(42,65,48,73,67,79,75,93,66,90,74,92,68,91,76,94)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=d,d^2=hi,e^2=g,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=fghi,f^c=fgh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42887);