Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #120 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12679 |
| prime radical | #585 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
618 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(1,2,6,13,7,14,32,51,8,15,33,52,35,54,81,102,9,16,34,53,36,55,82,103,5,12,29,48,30,49,78,100)(3,17,21,57,22,58,68,106,19,39,65,50,66,89,113,101,20,46,31,79,67,97,80,118,27,10,37,40,75,41,84,90)(4,11,24,43,25,44,72,94,23,42,69,91,70,92,114,123,26,45,73,95,74,96,116,125,28,47,76,98,77,99,117,126)(18,59,62,107,63,108,111,128,64,38,85,87,112,88,120,122,61,56,71,104,110,105,115,127,60,86,83,93,109,121,119,124);(1,3)(2,10)(4,18)(5,27)(6,31)(7,22)(8,19)(9,20)(11,38)(12,46)(13,50)(14,41)(15,17)(16,39)(21,34)(23,64)(24,71)(25,63)(26,61)(28,60)(29,65)(30,75)(32,80)(33,37)(35,66)(36,67)(40,53)(42,56)(43,93)(44,88)(45,86)(47,59)(48,57)(49,97)(51,101)(52,79)(54,58)(55,89)(62,73)(68,82)(69,83)(70,112)(72,115)(74,110)(76,85)(77,109)(78,113)(81,84)(87,95)(90,103)(91,107)(92,105)(94,124)(96,121)(98,104)(99,108)(100,106)(102,118)(111,116)(114,119)(117,120)(122,125)(123,128)(126,127);(1,4)(2,11)(3,18)(5,28)(6,24)(7,25)(8,23)(9,26)(10,38)(12,47)(13,43)(14,44)(15,42)(16,45)(17,56)(19,64)(20,61)(21,62)(22,63)(27,60)(29,76)(30,77)(31,71)(32,72)(33,69)(34,73)(35,70)(36,74)(37,83)(39,86)(40,87)(41,88)(46,59)(48,98)(49,99)(50,93)(51,94)(52,91)(53,95)(54,92)(55,96)(57,104)(58,105)(65,85)(66,112)(67,110)(68,111)(75,109)(78,117)(79,107)(80,115)(81,114)(82,116)(84,119)(89,121)(90,122)(97,108)(100,126)(101,124)(102,123)(103,125)(106,127)(113,120)(118,128)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^b=di,e^b=ei,f^b=fi,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59203);