Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2140 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12813 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,15,18,9)(4,21)(5,26)(6,33)(7,37)(8,11)(10,23)(12,30)(13,29)(14,35)(16,56,57,99)(17,61,27,83)(19,69,67,96)(20,42,68,55)(22,73)(24,78)(25,79)(28,51)(31,94)(32,80)(34,49)(36,97)(38,72)(39,58,53,76)(40,65,48,86)(41,64,54,85)(43,90)(44,89)(45,74)(46,93)(47,92)(50,111)(52,77)(59,118,116,128)(60,119,71,115)(62,82)(63,114,84,104)(66,124,87,120)(70,109,126,102)(75,112)(81,91)(88,108)(95,106)(98,107)(100,122,113,125)(101,121,105,117)(103,127,110,123);(1,3,7,20,8,18,35,68)(2,9,37,55,11,15,14,42)(4,16,38,71,22,57,25,60)(5,27,31,87,32,17,62,66)(6,19,29,64,36,67,92,85)(10,39,44,101,45,53,52,105)(12,48,81,103,46,40,50,110)(13,41,97,69,47,54,33,96)(21,56,72,115,73,99,79,119)(23,76,89,117,74,58,77,121)(24,59,98,126,75,116,34,70)(26,61,94,124,80,83,82,120)(28,84,95,122,88,63,90,125)(30,86,91,127,93,65,111,123)(43,100,51,104,106,113,108,114)(49,109,78,118,107,102,112,128);(1,4)(2,10)(3,16)(5,28)(6,34)(7,38)(8,22)(9,39)(11,45)(12,49)(13,51)(14,52)(15,53)(17,63)(18,57)(19,70)(20,71)(21,26)(23,30)(24,29)(25,35)(27,84)(31,95)(32,88)(33,43)(36,98)(37,44)(40,102)(41,104)(42,105)(46,107)(47,108)(48,109)(50,112)(54,114)(55,101)(56,61)(58,65)(59,64)(60,68)(62,90)(66,125)(67,126)(69,113)(72,94)(73,80)(74,93)(75,92)(76,86)(77,111)(78,81)(79,82)(83,99)(85,116)(87,122)(89,91)(96,100)(97,106)(103,118)(110,128)(115,124)(117,127)(119,120)(121,123)Polycyclic presentation:
2^9|a^2=f,b^2=dh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320367, #72320849, #72320882 |
| maximal normal subgroup of | #72320367, #72320849, #72320882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59483);