Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150878 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11886 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,16,19,63)(4,23)(5,31,34,61)(6,38)(7,41)(8,44)(9,15)(10,18)(11,28)(12,35,56,36)(13,25)(14,40)(17,65,67,113)(20,74,21,75)(22,76,45,64)(24,80,68,62)(26,86,33,100)(27,53)(29,94)(30,79)(32,58)(37,104,103,97)(39,107,42,54)(43,59)(46,81)(47,105)(48,98)(49,73)(50,70)(51,102,55,91)(52,84)(57,88,109,85)(60,72)(66,115)(69,122,82,120)(71,112)(77,89,96,90)(78,92)(83,118,95,116)(87,117)(93,108)(99,106)(101,114)(110,127)(111,121)(119,124)(123,126,128,125);(1,3,9,19)(2,10,15,49)(4,24,30,68)(5,32,34,71)(6,39,7,42)(8,45,40,22)(11,50,53,115)(12,54,56,107)(13,58,59,112)(14,60,44,48)(16,61,63,31)(17,28,67,27)(18,35,73,36)(20,25,21,43)(23,46,79,111)(26,87,33,101)(29,95,92,83)(37,105,103,99)(38,106,41,47)(51,116,55,118)(52,117,108,114)(57,75,109,74)(62,86,80,100)(64,97,76,104)(65,77,113,96)(66,91,70,102)(69,84,82,93)(72,88,98,85)(78,119,94,110)(81,89,121,90)(120,128,122,123)(124,126,127,125);(1,4,25,84,40,92,6,27,9,30,43,93,8,29,7,28)(2,11,41,94,44,108,59,79,15,53,38,78,14,52,13,23)(3,17,42,83,45,82,20,24,19,67,39,95,22,69,21,68)(5,33,85,126,103,91,35,90,34,26,88,125,37,102,36,89)(10,46,58,117,60,119,106,115,49,111,112,114,48,110,47,50)(12,55,104,123,109,86,61,77,56,51,97,128,57,100,31,96)(16,62,75,122,76,116,107,113,63,80,74,120,64,118,54,65)(18,70,105,127,98,101,71,121,73,66,99,124,72,87,32,81)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=ghi,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58076);