Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151643 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12430 |
| commutator, Frattini, cosocle | C4×C16 | C23 |
| prime radical | #135 | D8 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2170 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 44 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)(3,19,27,67)(4,28,32,85)(5,33,37,91)(6,15,40,59)(7,14,39,60)(8,17,9,16)(11,42,48,22)(12,49,52,30)(13,53,57,20)(21,71,75,118)(23,64,81,113)(24,63,80,114)(25,66,26,65)(29,86,88,107)(31,89,41,58)(34,55,70,56)(35,54,36,90)(38,61,94,51)(43,98,100,116)(44,97,103,68)(45,96,102,76)(46,77,47,78)(50,87,108,93)(62,105,111,73)(69,84,106,125)(72,83,95,124)(74,120,82,112)(79,115,123,110)(92,126,117,109)(99,121,104,128)(101,119,127,122);(1,3,20,68,7,24,35,77,8,25,70,96,40,81,37,42,10,27,53,97,39,80,36,78,9,26,34,76,6,23,5,22)(2,11,33,64,15,45,55,65,16,46,90,114,60,103,57,67,18,48,91,113,59,102,56,66,17,47,54,63,14,44,13,19)(4,21,69,116,30,73,93,122,41,82,117,128,94,123,88,95,32,75,106,98,49,105,87,119,31,74,92,121,38,79,29,72)(12,43,84,71,28,83,86,115,61,104,126,120,89,127,108,111,52,100,125,118,85,124,107,110,51,99,109,112,58,101,50,62);(1,4)(2,12)(3,21)(5,29)(6,38)(7,30)(8,41)(9,31)(10,32)(11,43)(13,50)(14,58)(15,28)(16,61)(17,51)(18,52)(19,62)(20,69)(22,72)(23,79)(24,73)(25,82)(26,74)(27,75)(33,84)(34,92)(35,93)(36,87)(37,88)(39,49)(40,94)(42,95)(44,101)(45,83)(46,104)(47,99)(48,100)(53,106)(54,109)(55,86)(56,107)(57,108)(59,85)(60,89)(63,112)(64,71)(65,115)(66,110)(67,111)(68,116)(70,117)(76,121)(77,122)(78,119)(80,105)(81,123)(90,126)(91,125)(96,128)(97,98)(102,124)(103,127)(113,118)(114,120)Polycyclic presentation:
2^9|a^2=i,b^2=df,d^2=fh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^c=eg,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58840);