Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13493 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,5,12,29,50,77,70,8,13,32,51,38,56,89,111,9,15,33,53,36,90,88,124,6,34,30,86,7,14,31,52)(3,10,17,40,62,98,116,113,18,43,63,101,68,105,120,123,20,44,65,102,91,103,126,128,35,41,87,99,19,42,64,100)(4,21,24,55,37,92,85,47,27,72,79,107,76,109,83,46,23,71,75,112,81,110,84,54,25,80,78,48,26,49,28,11)(16,57,59,104,67,122,82,96,22,73,74,125,93,127,117,94,58,114,115,106,121,108,119,45,66,69,118,95,60,97,61,39);(1,3)(2,10)(4,22)(5,17)(6,35)(7,19)(8,18)(9,20)(11,45)(12,40)(13,43)(14,42)(15,44)(16,25)(21,69)(23,66)(24,74)(26,67)(27,58)(28,82)(29,62)(30,87)(31,64)(32,63)(33,65)(34,41)(36,91)(37,93)(38,68)(39,47)(46,96)(48,106)(49,108)(50,98)(51,101)(52,100)(53,102)(54,94)(55,95)(56,105)(57,72)(59,78)(60,81)(61,84)(70,113)(71,73)(75,118)(76,121)(77,116)(79,115)(80,114)(83,119)(85,117)(86,99)(88,126)(89,120)(90,103)(92,97)(104,107)(109,122)(110,127)(111,123)(112,125)(124,128);(1,4,8,25,9,23,6,27)(2,11,13,54,15,46,34,47)(3,16,18,66,20,58,35,22)(5,28,32,84,33,83,30,85)(7,37,29,26,38,81,36,76)(10,39,43,45,44,94,41,96)(12,49,51,110,53,109,86,92)(14,55,50,48,56,112,90,107)(17,61,63,119,65,117,87,82)(19,67,62,60,68,121,91,93)(21,70,80,111,71,124,72,52)(24,77,78,89,75,88,79,31)(40,97,101,108,102,127,99,122)(42,104,98,95,105,106,103,125)(57,113,69,123,114,128,73,100)(59,116,118,120,115,126,74,64)Polycyclic presentation:
2^9|a^2=e,c^2=dh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320831, #72320903 |
| maximal normal subgroup of | #72320516, #72320831, #72320903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60569);