Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143170 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10026 |
| commutator, Frattini, cosocle | #405 | C23 |
| prime radical | #575 | D4 |
| LCS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,10,18)(3,19,24,74)(4,25,32,69)(5,33,40,64)(6,41,43,15)(7,14,42,67)(8,17,9,16)(11,47,52,21)(12,53,59,28)(13,39,65,38)(20,75,78,120)(22,82,84,125)(23,85,46,60)(26,72,91,127)(27,92,97,121)(29,57,98,58)(30,66,31,56)(34,71,104,51)(35,105,96,88)(36,62,108,124)(37,63,45,70)(44,73,89,117)(48,111,114,93)(49,103,50,102)(54,79,113,110)(55,99,101,95)(61,94,122,107)(68,80,109,81)(76,115,128,116)(77,123,86,119)(83,112,100,126)(87,106,118,90);(1,3)(2,11)(4,26)(5,34)(6,22)(7,44)(8,46)(9,23)(10,24)(12,54)(13,60)(14,49)(15,68)(16,71)(17,51)(18,52)(19,33)(20,29)(21,38)(25,87)(27,93)(28,86)(30,100)(31,83)(32,91)(35,106)(36,102)(37,80)(39,47)(40,104)(41,109)(42,89)(43,84)(45,81)(48,56)(50,67)(53,77)(55,120)(57,115)(58,116)(59,113)(61,123)(62,82)(63,117)(64,74)(65,85)(66,114)(69,118)(70,73)(72,92)(75,101)(76,94)(78,98)(79,99)(88,126)(90,96)(95,110)(97,111)(103,108)(105,112)(107,128)(119,122)(121,127)(124,125);(1,4,10,32)(2,12,18,59)(3,20,24,78)(5,35,40,96)(6,29,43,98)(7,28,42,53)(8,31,9,30)(11,48,52,114)(13,61,65,122)(14,66,67,56)(15,69,41,25)(16,58,17,57)(19,72,74,127)(21,79,47,110)(22,83,84,100)(23,86,46,77)(26,89,91,44)(27,38,97,39)(33,101,64,55)(34,76,104,128)(36,107,108,94)(37,95,45,99)(49,115,50,116)(51,118,71,87)(54,109,113,68)(60,112,85,126)(62,105,124,88)(63,121,70,92)(73,119,117,123)(75,125,120,82)(80,106,81,90)(93,103,111,102)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fg,g^c=gi,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50368);