Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2158 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12907 |
| prime radical | C23 | #508 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #117 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 180 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)(4,24,27,75)(5,30)(6,13)(7,36)(8,38)(9,15)(10,19)(11,26,44,29)(12,17)(14,31)(18,59,62,110)(20,51)(21,70)(22,72)(23,53)(25,77,80,123)(28,84,83,45)(32,86)(33,89)(34,88)(35,48)(37,90)(39,92,93,114)(40,66)(41,95)(42,68)(43,97,98,108)(46,56)(47,58)(49,61,100,64)(50,54)(52,65)(55,74,107,120)(57,87)(60,73,113,119)(63,117,116,101)(67,91)(69,103)(71,118)(76,82,121,78)(79,122,81,85)(94,104,109,127)(96,105)(99,124,126,125)(102,106)(111,115,128,112);(1,3,17,54,7,21,32,66,6,20,56,105,35,69,34,68,9,23,58,106,37,71,33,67,8,22,57,95,31,65,5,19)(2,10,30,52,14,41,87,72,38,91,89,118,90,102,47,53,15,42,88,103,48,96,46,51,13,40,86,70,36,50,12,16)(4,18,55,104,28,63,79,92,26,61,97,124,82,115,80,113,27,62,107,127,83,116,81,114,29,64,108,125,78,112,25,60)(11,39,85,101,45,94,120,110,75,119,123,128,121,126,98,100,44,93,122,117,84,109,74,59,24,73,77,111,76,99,43,49);(1,4)(2,11,15,44)(3,18)(5,25)(6,26)(7,28)(8,29)(9,27)(10,39,42,93)(12,43,47,98)(13,24,38,75)(14,45,48,84)(16,49,53,100)(17,55)(19,60)(20,61)(21,63)(22,64)(23,62)(30,85,88,122)(31,78)(32,79)(33,81)(34,80)(35,82)(36,76,90,121)(37,83)(40,73,91,119)(41,94,96,109)(46,74,87,120)(50,99,102,126)(51,59,72,110)(52,101,103,117)(54,104)(56,97)(57,108)(58,107)(65,112)(66,92)(67,114)(68,113)(69,115)(70,111,118,128)(71,116)(77,89,123,86)(95,125)(105,124)(106,127)Polycyclic presentation:
2^9|a^2=f,b^2=dg,c^2=eh,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320480, #72320842, #72320882, … |
| maximal normal subgroup of | #72320480, #72320842, #72320882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59605);