Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144222 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10320 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | #585 | D4 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 554 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:
(1,2,8,14)(3,15,20,57)(4,21,25,68)(5,26,31,54)(6,12,33,51)(7,35,28,13)(9,37,41,17)(10,42,45,23)(11,46,49,30)(16,58,62,116)(18,56,32,87)(19,65,36,47)(22,70,75,114)(24,77,72,69)(27,53,82,40)(29,80,85,112)(34,91,83,52)(38,94,96,118)(39,89,50,63)(43,100,102,126)(44,76,92,99)(48,105,108,86)(55,88,113,60)(59,66,93,109)(61,120,78,117)(64,107,90,106)(67,104,123,74)(71,115,125,122)(73,79,98,111)(81,110,84,97)(95,124,103,128)(101,119,121,127);(1,3,8,20)(2,9,14,41)(4,16,25,62)(5,27,31,82)(6,32,33,18)(7,36,28,19)(10,38,45,96)(11,47,49,65)(12,50,51,39)(13,53,35,40)(15,54,57,26)(17,46,37,30)(21,66,68,109)(22,71,75,125)(23,60,42,88)(24,78,72,61)(29,84,85,81)(34,64,83,90)(43,101,102,121)(44,103,92,95)(48,107,108,106)(52,97,91,110)(55,111,113,79)(56,80,87,112)(58,114,116,70)(59,104,93,74)(63,86,89,105)(67,117,123,120)(69,115,77,122)(73,124,98,128)(76,127,99,119)(94,126,118,100);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,38)(11,43)(12,21)(13,44)(14,45)(15,55)(17,59)(18,60)(19,61)(20,62)(26,79)(27,71)(28,72)(29,73)(30,74)(31,75)(32,88)(33,42)(34,76)(35,92)(36,78)(37,93)(39,66)(40,95)(41,96)(46,104)(47,101)(48,67)(49,102)(50,109)(51,68)(52,69)(53,103)(54,111)(56,58)(57,113)(63,118)(64,119)(65,121)(70,80)(77,91)(81,124)(82,125)(83,99)(84,128)(85,98)(86,126)(87,116)(89,94)(90,127)(97,122)(100,105)(106,117)(107,120)(108,123)(110,115)(112,114)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51420);