Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C2×#1415 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,12,9,15,8,40)(3,16,19,60,17,59,41,62)(4,22,25,70,29,73,28,88)(6,32,30,90,36,93,35,96)(7,37,31,58,39,98,34,14)(10,42,44,63,43,101,47,18)(11,48,50,84,52,83,100,26)(13,54,53,80,56,79,92,24)(20,68,67,121,64,119,95,112)(21,69,38,61,65,118,99,117)(23,55,75,114,74,115,89,116)(27,85,81,123,87,127,82,72)(33,49,91,110,94,109,97,111)(45,86,104,125,102,128,106,77)(46,107,105,124,103,66,57,76)(51,78,71,108,113,126,122,120);(1,3)(2,10)(4,23)(5,19)(6,33)(7,38)(8,41)(9,17)(11,49)(12,44)(13,55)(14,57)(15,43)(16,22)(18,26)(20,24)(21,34)(25,75)(27,86)(28,89)(29,74)(30,91)(31,65)(32,45)(35,97)(36,94)(37,46)(39,99)(40,47)(42,48)(50,110)(51,112)(52,109)(53,114)(54,67)(56,115)(58,105)(59,73)(60,70)(61,85)(62,88)(63,84)(64,80)(66,120)(68,71)(69,72)(76,78)(77,82)(79,95)(81,125)(83,101)(87,128)(90,104)(92,116)(93,102)(96,106)(98,103)(100,111)(107,108)(113,121)(117,127)(118,123)(119,122)(124,126);(3,17)(10,43)(16,59)(18,63)(19,41)(20,64)(21,65)(23,74)(33,94)(38,99)(42,101)(44,47)(45,102)(46,103)(49,109)(55,115)(57,105)(60,62)(61,117)(66,107)(67,95)(68,119)(69,118)(75,89)(76,124)(77,125)(86,128)(91,97)(104,106)(110,111)(112,121)(114,116)Polycyclic presentation:
2^9|a^2=e,d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=df,f^b=fg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321140, #72321197, #72321207, … |
| maximal normal subgroup of | #72321140, #72321197, #72321207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60756);