Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13476 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,11,18,10,19,9,50)(3,20,26,98,27,81,52,80)(4,28,34,68,35,100,51,64)(5,16,53,55,39,74,38,15)(6,40,37,14,44,70,7,45)(8,48,36,73,47,69,43,17)(12,54,79,22,61,90,60,91)(13,32,76,102,67,111,66,31)(21,82,99,124,87,123,86,117)(23,59,89,78,41,112,24,93)(25,57,97,58,95,71,88,121)(29,101,113,62,106,116,84,77)(30,107,109,127,46,75,42,65)(33,72,49,63,110,114,108,126)(56,85,118,103,119,104,122,96)(83,115,105,125,94,128,92,120);(1,3,6,41,8,25,5,22,11,52,37,89,36,88,53,54,10,27,44,23,47,95,39,91,9,26,7,24,43,97,38,90)(2,12,55,57,17,59,14,20,50,79,16,58,69,78,40,98,19,61,15,71,73,112,45,81,18,60,74,121,48,93,70,80)(4,29,102,103,33,105,30,87,51,113,32,104,108,94,42,86,35,106,31,96,110,92,46,21,34,84,111,85,49,83,109,99)(13,62,28,82,65,125,63,56,76,101,68,117,107,115,114,122,67,77,100,123,127,120,126,119,66,116,64,124,75,128,72,118);(1,4)(2,13)(3,21)(5,32)(6,42)(7,46)(8,49)(9,51)(10,35)(11,34)(12,56)(14,68)(15,72)(16,63)(17,75)(18,76)(19,67)(20,77)(22,29)(23,92)(24,94)(25,96)(26,99)(27,87)(28,40)(30,37)(31,38)(33,43)(36,110)(39,111)(41,105)(44,109)(45,64)(47,108)(48,65)(50,66)(52,86)(53,102)(54,84)(55,114)(57,120)(58,115)(59,82)(60,122)(61,119)(62,81)(69,127)(70,100)(71,125)(73,107)(74,126)(78,124)(79,118)(80,116)(83,89)(85,97)(88,104)(90,113)(91,106)(93,117)(95,103)(98,101)(112,123)(121,128)Polycyclic presentation:
2^9|a^2=de,b^2=dgh,d^2=ghi,e^2=g,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=egh,f^c=fghi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320842, #72320873 |
| maximal normal subgroup of | #72320516, #72320842, #72320873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60546);