Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138944 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9259 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#578 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,7,14,42,16,39,13,9,15,38,61,8,43,6,37)(3,17,22,67,24,87,21,82,23,68,84,117,79,56,40,66)(4,25,28,54,44,60,97,64,30,62,96,11,29,53,27,55)(5,31,34,63,93,101,18,46,36,12,41,58,35,59,71,100)(10,45,49,81,108,86,48,103,51,69,107,20,50,80,115,32)(19,72,75,65,77,121,74,110,76,126,83,123,88,125,85,122)(26,91,94,119,106,120,70,57,95,90,98,52,33,104,105,99)(47,92,89,109,118,102,111,128,113,78,114,124,112,73,116,127);(1,3,18,69)(2,10,46,82)(4,19,70,124)(5,32,7,40)(6,22,35,81)(8,24,34,80)(9,23,71,45)(11,47,90,125)(12,56,61,115)(13,49,101,17)(14,48,31,87)(15,51,100,66)(16,50,58,67)(20,39,84,93)(21,36,103,38)(25,89,57,110)(26,92,28,85)(27,75,33,102)(29,77,94,127)(30,76,105,109)(37,107,59,68)(41,86,42,79)(43,108,63,117)(44,88,98,128)(52,65,60,116)(53,111,119,123)(54,113,91,121)(55,112,104,126)(62,114,99,122)(64,118,120,72)(73,97,83,106)(74,95,78,96);(1,4,9,30)(2,11,15,54)(3,19,23,76)(5,33,36,106)(6,28,39,96)(7,27,38,97)(8,44,42,29)(10,47,51,113)(12,57,31,99)(13,60,37,53)(14,62,61,25)(16,64,43,55)(17,65,68,123)(18,70,71,105)(20,78,81,92)(21,83,40,75)(22,85,84,74)(24,88,79,77)(26,93,95,35)(32,102,103,73)(34,98,41,94)(45,109,69,124)(46,90,100,91)(48,114,115,89)(49,116,107,111)(50,118,108,112)(52,59,119,101)(56,110,87,122)(58,120,63,104)(66,121,82,125)(67,72,117,126)(80,128,86,127)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=i,d^2=gh,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46147);