Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| 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:
(1,2,6,13,35,56,96,118,8,15,37,58,40,61,97,119,9,16,38,59,32,53,90,113,5,12,31,52,7,14,36,57)(3,17,20,63,71,107,127,115,19,45,70,54,75,112,93,123,22,51,33,91,88,44,101,105,30,10,41,43,21,64,73,122)(4,23,26,60,39,98,89,50,29,79,86,111,87,117,95,48,27,78,85,110,84,116,94,47,25,77,83,49,28,55,34,11)(18,65,67,103,74,104,100,42,66,76,124,106,82,128,92,102,24,80,81,109,99,114,126,46,69,108,125,120,68,121,72,62);(1,3)(2,10)(4,24)(5,30)(6,33)(7,21)(8,19)(9,22)(11,46)(12,51)(13,54)(14,44)(15,17)(16,45)(18,27)(20,38)(23,76)(25,66)(26,67)(28,82)(29,69)(31,70)(32,88)(34,72)(35,71)(36,93)(37,41)(39,99)(40,75)(42,48)(43,59)(47,102)(49,109)(50,62)(52,63)(53,112)(55,104)(56,64)(57,115)(58,91)(60,120)(61,107)(65,77)(68,87)(73,97)(74,84)(78,108)(79,80)(81,85)(83,125)(86,124)(89,100)(90,127)(92,95)(94,126)(96,101)(98,128)(103,111)(105,119)(106,110)(113,122)(114,117)(116,121)(118,123);(1,4)(2,11)(3,18,22,24)(5,29)(6,34)(7,39)(8,25)(9,27)(10,42,45,46)(12,50)(13,55)(14,60)(15,47)(16,48)(17,62,51,102)(19,69,30,66)(20,72,33,92)(21,74,75,99)(23,57)(26,36)(28,35)(31,89)(32,87)(37,94)(38,95)(40,84)(41,100,70,126)(43,104,54,114)(44,106,107,120)(49,56)(52,98)(53,111)(58,116)(59,117)(61,110)(63,121,91,128)(64,103,112,109)(65,123,80,122)(67,93,81,73)(68,88,82,71)(76,115,108,105)(77,118)(78,119)(79,113)(83,96)(85,97)(86,90)(101,124,127,125)Polycyclic presentation:
2^9|a^2=e,c^2=dh,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60570);