Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1339 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| socle | C2 | #7081 |
| prime radical | C23 | #362 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,13,9,48,39,104,10,17,40,67,8,16,38,66)(3,18,20,73,49,76,100,125,24,77,84,112,23,75,34,99)(4,25,28,88,32,98,96,128,33,70,35,101,31,60,95,120)(6,41,44,105,45,14,68,71,27,15,36,65,7,46,37,64)(11,50,51,19,55,78,63,108,56,111,116,81,54,110,115,80)(12,57,59,30,89,94,127,43,62,93,122,97,61,103,121,29)(21,85,87,126,22,74,83,119,47,58,82,118,42,106,107,124)(26,53,90,114,79,52,92,117,91,69,109,123,86,72,102,113);(1,3,8,23,10,24,9,49)(2,11,16,54,17,56,48,55)(4,26,31,86,33,91,32,79)(5,34,38,84,40,100,39,20)(6,42,7,47,27,22,45,21)(12,58,61,74,62,85,89,106)(13,63,66,51,67,115,104,116)(14,69,41,52,46,53,15,72)(18,70,75,98,77,25,76,60)(19,29,80,97,81,43,108,30)(28,92,95,90,35,102,96,109)(36,87,68,107,44,82,37,83)(50,93,110,94,111,57,78,103)(59,119,121,126,122,124,127,118)(64,117,65,114,71,113,105,123)(73,120,99,101,112,128,125,88)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fhi,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,h^b=hi,d^c=dghi,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1015);