Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512929 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2206 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #33981 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 128 points:
(1,2,8,14)(3,15,17,56)(4,21,26,77)(5,27,30,11)(6,31,34,91)(7,35,37,84)(9,38,40,18)(10,43,47,23)(12,49,51,33)(13,52,54,29)(16,59,60,115)(19,69,65,120)(20,72,66,42)(22,76,25,75)(24,81,82,107)(28,85,88,90)(32,53,93,105)(36,50,79,110)(39,96,97,118)(41,101,100,123)(44,78,46,80)(45,86,106,92)(48,87,83,109)(55,104,99,62)(57,108,112,71)(58,95,113,68)(61,74,64,116)(63,102,117,128)(67,89,94,124)(70,98,121,127)(73,114,119,126)(103,125,111,122);(1,3,5,18,8,17,30,38)(2,9,11,15,14,40,27,56)(4,16,22,61,26,60,25,64)(6,19,28,67,34,65,88,94)(7,20,29,68,37,66,52,95)(10,39,44,55,47,97,46,99)(12,41,48,57,51,100,83,112)(13,42,35,58,54,72,84,113)(21,74,75,59,77,116,76,115)(23,62,78,118,43,104,80,96)(24,63,79,119,82,117,36,73)(31,89,90,69,91,124,85,120)(32,70,86,122,93,121,92,125)(33,71,87,123,49,108,109,101)(45,98,105,111,106,127,53,103)(50,102,107,114,110,128,81,126);(1,4)(2,10)(3,16)(5,22)(6,32)(7,36)(8,26)(9,39)(11,44)(12,50)(13,53)(14,47)(15,55)(17,60)(18,61)(19,70)(20,73)(21,31)(23,33)(24,29)(25,30)(27,46)(28,86)(34,93)(35,45)(37,79)(38,64)(40,97)(41,102)(42,103)(43,49)(48,107)(51,110)(52,82)(54,105)(56,99)(57,114)(58,98)(59,69)(62,71)(63,68)(65,121)(66,119)(67,122)(72,111)(74,89)(75,90)(76,85)(77,91)(78,87)(80,109)(81,83)(84,106)(88,92)(94,125)(95,117)(96,101)(100,128)(104,108)(112,126)(113,127)(115,120)(116,124)(118,123);(3,17)(9,40)(15,56)(16,60)(18,38)(19,65)(20,66)(39,97)(41,100)(42,72)(55,99)(57,112)(58,113)(59,115)(61,64)(62,104)(63,117)(67,94)(68,95)(69,120)(70,121)(71,108)(73,119)(74,116)(89,124)(96,118)(98,127)(101,123)(102,128)(103,111)(114,126)(122,125)Polycyclic presentation:
2^9|a^2=i,b^2=e,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^c=fg,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420126);