Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142616 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9946 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,8,13)(3,15,21,65)(4,22,26,82)(5,27,32,89)(6,34,38,80)(7,39,30,14)(9,41,44,17)(10,47,50,24)(11,52,56,33)(12,57,60,37)(16,66,71,124)(18,75,78,122)(19,64,20,53)(23,83,86,127)(25,87,40,59)(28,46,74,45)(29,81,93,118)(31,94,97,55)(35,63,77,117)(36,61,99,51)(42,100,48,110)(43,105,107,91)(49,113,114,92)(54,96,116,85)(58,95,108,73)(62,119,120,68)(67,79,90,106)(69,123,70,121)(72,102,84,112)(76,111,109,101)(88,115,126,98)(103,125,104,128);(1,3)(2,9)(4,16)(5,28)(6,35)(7,20)(8,21)(10,48)(11,53)(12,58)(13,44)(14,46)(15,27)(17,33)(18,37)(19,30)(22,79)(23,84)(24,76)(25,70)(26,71)(29,67)(31,95)(32,74)(34,43)(36,68)(38,77)(39,45)(40,69)(41,52)(42,50)(47,109)(49,62)(51,103)(54,111)(55,117)(56,64)(57,78)(59,112)(60,108)(61,104)(63,94)(65,89)(66,83)(72,86)(73,97)(75,88)(80,107)(81,121)(82,106)(85,128)(87,102)(90,93)(91,98)(92,100)(96,125)(99,119)(101,116)(105,115)(110,113)(114,120)(118,123)(122,126)(124,127);(1,4,8,26)(2,10,13,50)(3,16,21,71)(5,29,32,93)(6,36,38,99)(7,40,30,25)(9,42,44,48)(11,54,56,116)(12,59,60,87)(14,61,39,51)(15,62,65,120)(17,72,41,84)(18,76,78,109)(19,70,20,69)(22,80,82,34)(23,52,86,33)(24,57,47,37)(27,49,89,114)(28,90,74,67)(31,96,97,85)(35,68,77,119)(43,106,107,79)(45,104,46,103)(53,111,64,101)(55,118,94,81)(58,102,108,112)(63,121,117,123)(66,122,124,75)(73,125,95,128)(83,126,127,88)(91,110,105,100)(92,115,113,98)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,d^c=dh,e^c=eg,g^c=gi,h^c=hi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49815);