Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C2×M16 | C2×C8 |
| Frattini, cosocle | #2151 | C22 |
| socle | C2 | #7066 |
| prime radical | C23 | #350 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,11,6,12,31,46,8,13,32,47,34,48,85,102,7,14,30,45,33,49,84,100,23,50,72,101,73,103,117,127)(3,15,17,52,19,53,59,105,20,40,60,93,64,94,110,124,18,54,61,88,62,106,111,122,63,65,28,79,80,112,81,120)(4,21,24,66,26,67,75,113,27,68,76,114,78,115,119,128,25,10,29,41,77,42,82,95,55,43,83,96,107,97,121,125)(9,35,36,86,38,87,89,16,39,56,44,57,92,108,98,22,37,69,90,70,91,116,123,58,51,74,99,109,104,118,126,71);(1,3,7,18)(2,9)(4,22)(5,28)(6,19,33,62)(8,20,23,63)(10,40,21,65)(11,44,45,99)(12,38)(13,39)(14,37)(15,43,54,68)(16,55)(17,32)(24,74,29,56)(25,71)(26,70)(27,58)(30,60)(31,81)(34,64,73,80)(35,83,69,76)(36,47,90,101)(41,52)(42,94,67,112)(46,98,100,126)(48,92)(49,91)(50,51)(53,97,106,115)(57,107)(59,85)(61,72)(66,88)(75,118,82,108)(77,86)(78,109)(79,114)(84,110)(87,121,116,119)(89,102,123,127)(93,96)(95,105)(103,104)(111,117)(113,122)(120,128)(124,125)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=gh,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=deghi,e^b=ei,g^b=gi,d^c=dg,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 937);