Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1835 |
| prime radical | C23 | #427 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,31)(20,38,42,69)(21,44)(22,27)(23,47)(24,30)(25,34)(26,41,54,75)(28,32)(29,45)(33,52,39,48)(35,58)(36,49)(37,60)(40,73,62,55)(43,76,74,53)(46,56)(50,67)(51,61)(57,64,79,71)(59,66)(63,77,70,68)(65,78,72,80);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,32,61,45,66,22,35,24,37,21,34,23,36,46,67)(18,25,27,31,47,58,28,49,30,51,56,60,29,50,44,59)(20,39,62,63,74,65,41,71,42,33,40,70,43,72,75,64)(26,52,53,79,73,78,38,68,54,48,76,57,55,80,69,77);(3,9)(6,12)(10,15)(13,16)(17,20,21,40,45,74,46,75,24,42,32,62,23,43,22,41)(18,26,27,53,47,73,28,38,30,54,56,76,29,55,44,69)(19,33,34,63,66,72,67,71,37,39,61,70,36,65,35,64)(25,48,31,57,58,80,49,77,51,52,60,79,50,78,59,68)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=chi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42876);