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 | |
497 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 497 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:
(1,2,6,13,32,14,65,64,7,15,39,12,8,41,40,60)(3,16,21,69,22,89,88,120,23,70,20,82,84,53,37,103)(4,24,27,51,42,62,38,52,28,93,26,66,29,59,98,10)(5,11,34,55,85,56,73,44,35,57,33,63,36,102,17,71)(9,43,47,81,68,87,61,99,48,106,46,19,49,80,58,30)(18,74,76,108,77,126,86,67,78,121,104,117,90,122,83,114)(25,92,94,118,31,100,101,50,95,115,97,54,96,107,72,116)(45,75,109,124,113,79,91,105,110,125,119,123,111,128,112,127);(1,3,17,43,7,23,73,106)(2,9,44,103,15,48,71,120)(4,18,72,105,28,78,101,127)(5,30,6,37,35,99,39,88)(8,22,33,87,32,84,34,80)(10,45,107,74,52,110,100,121)(11,53,12,58,57,89,13,61)(14,49,63,82,41,68,55,69)(16,64,47,56,70,60,46,102)(19,65,20,36,81,40,21,85)(24,91,50,114,93,112,116,67)(25,79,27,83,95,128,26,86)(29,77,97,123,42,90,94,124)(31,75,38,104,96,125,98,76)(51,111,92,122,66,113,115,126)(54,117,59,119,118,108,62,109);(1,4)(2,10)(3,18)(5,31)(6,38)(7,28)(8,42)(9,45)(11,54)(12,59)(13,62)(14,66)(15,52)(16,67)(17,72)(19,79)(20,83)(21,86)(22,90)(23,78)(24,64)(25,85)(26,40)(27,65)(29,32)(30,75)(33,94)(34,97)(35,96)(36,95)(37,104)(39,98)(41,51)(43,105)(44,107)(46,112)(47,91)(48,110)(49,113)(50,56)(53,117)(55,92)(57,118)(58,119)(60,93)(61,109)(63,115)(68,111)(69,122)(70,114)(71,100)(73,101)(74,103)(76,88)(77,84)(80,123)(81,128)(82,126)(87,124)(89,108)(99,125)(102,116)(106,127)(120,121)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gh,e^2=gh,f^2=gh,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, 46140);