Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144240 |
| 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 | |
552 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 552 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)(3,15,20,58)(4,21,25,52)(5,26,31,55)(6,32,35,12)(7,37,28,13)(9,39,43,17)(10,34,46,23)(11,47,50,30)(16,59,63,115)(18,65,33,57)(19,68,38,48)(22,72,77,116)(24,79,74,71)(27,54,84,42)(29,86,89,82)(36,53,85,92)(40,98,100,119)(41,64,51,93)(44,103,105,127)(45,78,96,95)(49,90,91,108)(56,66,114,61)(60,69,97,101)(62,121,80,118)(67,109,94,110)(70,107,111,76)(73,117,126,124)(75,81,88,113)(83,102,87,112)(99,125,106,128)(104,120,123,122);(1,3,8,20)(2,9,14,43)(4,16,25,63)(5,27,31,84)(6,33,35,18)(7,38,28,19)(10,40,46,100)(11,48,50,68)(12,51,32,41)(13,54,37,42)(15,55,58,26)(17,47,39,30)(21,69,52,101)(22,73,77,126)(23,61,34,66)(24,80,74,62)(29,87,89,83)(36,67,85,94)(44,104,105,123)(45,106,96,99)(49,110,91,109)(53,102,92,112)(56,113,114,81)(57,82,65,86)(59,116,115,72)(60,107,97,76)(64,90,93,108)(70,118,111,121)(71,117,79,124)(75,125,88,128)(78,122,95,120)(98,127,119,103);(1,4)(2,10)(3,16)(5,22)(6,34)(7,24)(8,25)(9,40)(11,44)(12,52)(13,45)(14,46)(15,56)(17,60)(18,66)(19,62)(20,63)(21,32)(23,35)(26,81)(27,73)(28,74)(29,88)(30,76)(31,77)(33,61)(36,95)(37,96)(38,80)(39,97)(41,101)(42,99)(43,100)(47,107)(48,104)(49,111)(50,105)(51,69)(53,79)(54,106)(55,113)(57,115)(58,114)(59,65)(64,98)(67,122)(68,123)(70,91)(71,92)(72,86)(75,89)(78,85)(82,116)(83,128)(84,126)(87,125)(90,103)(93,119)(94,120)(102,117)(108,127)(109,121)(110,118)(112,124)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,e^c=ei,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, 51439);