Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152176 |
| 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 | |
609 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 609 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)(3,16,20,60)(4,22,25,49)(5,27)(6,13)(7,35)(8,39)(9,15)(10,18,43,72)(11,24,47,34)(12,31)(14,37)(17,63)(19,59,40,62)(21,75,74,110)(23,79,83,116)(26,48,86,108)(28,54,73,58)(29,55)(30,89)(32,95)(33,56)(36,61,70,50)(38,51)(41,99)(42,71,45,98)(44,96,103,91)(46,104,106,128)(52,92)(53,69)(57,67)(64,102)(65,114)(66,105)(68,115)(76,122)(77,118,107,84)(78,81,97,85)(80,125)(82,111,93,88)(87,117)(90,94)(100,120)(101,121)(109,127)(112,126)(113,123)(119,124);(1,3)(2,10)(4,17)(5,28)(6,19)(7,36)(8,40)(9,20)(11,41)(12,50)(13,42)(14,54)(15,43)(16,56)(18,69)(21,37)(22,76)(23,80)(24,65)(25,68)(26,87)(27,60)(29,74)(30,91)(31,72)(32,96)(33,73)(34,67)(35,44)(38,70)(39,45)(46,105)(47,101)(48,109)(49,102)(51,103)(52,75)(53,61)(55,58)(57,111)(59,95)(62,89)(63,116)(64,118)(66,85)(71,94)(77,124)(78,125)(79,115)(81,123)(82,126)(83,120)(84,122)(86,119)(88,114)(90,110)(92,98)(93,127)(97,100)(99,128)(104,121)(106,113)(107,117)(108,112);(1,4,9,25)(2,11,15,47)(3,17,20,68)(5,23,33,83)(6,34,8,24)(7,26,38,86)(10,41,43,101)(12,46,53,106)(13,49,39,22)(14,48,55,108)(16,57,60,114)(18,64,72,122)(19,67,40,65)(21,66,74,123)(27,88,56,111)(28,80,73,120)(29,81,37,85)(30,93,32,82)(31,84,69,118)(35,97,51,78)(36,87,70,119)(42,102,45,76)(44,100,103,125)(50,105,61,113)(52,107,90,77)(54,109,58,112)(59,115,62,63)(71,121,98,99)(75,117,110,124)(79,89,116,95)(91,127,96,126)(92,128,94,104)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59373);