Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152169 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12731 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,6,13,8,12,35,15)(3,16,19,59,22,58,21,71)(4,23,25,53,28,10,27,45)(5,29,32,88,34,87,94,89)(7,36,31,14,38,54,93,52)(9,39,42,66,41,65,56,18)(11,47,50,95,49,106,51,33)(17,60,62,112,64,57,67,111)(20,69,68,107,70,109,37,48)(24,75,78,123,81,86,80,127)(26,83,82,46,85,96,77,74)(30,55,90,101,92,110,91,43)(40,97,98,61,72,115,100,116)(44,102,104,79,73,122,108,126)(63,117,84,114,118,120,119,103)(76,113,124,121,125,105,128,99);(1,3)(2,9)(4,17)(5,30)(6,19)(7,37)(8,22)(10,40)(11,48)(12,41)(13,42)(14,55)(15,56)(16,29)(18,33)(20,31)(21,35)(23,72)(24,76)(25,62)(26,84)(27,67)(28,64)(32,90)(34,92)(36,43)(38,68)(39,47)(44,103)(45,98)(46,105)(49,107)(50,69)(51,109)(52,110)(53,100)(54,101)(57,86)(58,87)(59,88)(60,75)(61,79)(63,77)(65,106)(66,95)(70,93)(71,89)(73,114)(74,113)(78,124)(80,128)(81,125)(82,118)(83,121)(85,119)(91,94)(96,99)(97,102)(104,117)(108,120)(111,127)(112,123)(115,122)(116,126);(1,4,8,28)(2,10,12,23)(3,17,22,64)(5,24,34,81)(6,27,35,25)(7,26,38,85)(9,40,41,72)(11,44,49,73)(13,53,15,45)(14,46,52,74)(16,57,58,60)(18,61,66,116)(19,67,21,62)(20,63,70,118)(29,86,87,75)(30,76,92,125)(31,77,93,82)(32,80,94,78)(33,79,95,126)(36,96,54,83)(37,84,68,119)(39,97,65,115)(42,100,56,98)(43,99,101,121)(47,102,106,122)(48,103,107,114)(50,108,51,104)(55,105,110,113)(59,112,71,111)(69,120,109,117)(88,123,89,127)(90,128,91,124)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59368);