Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1339 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| socle | C2 | #7080 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 128 points:
(1,2,5,13,9,48,39,104,10,17,40,67,8,16,38,66)(3,18,20,73,49,76,100,125,24,77,84,112,23,75,34,99)(4,25,28,88,32,98,96,128,33,70,35,101,31,60,95,120)(6,41,44,105,45,14,68,71,27,15,36,65,7,46,37,64)(11,50,51,19,55,78,63,108,56,111,116,81,54,110,115,80)(12,57,59,30,89,94,127,43,62,93,122,97,61,103,121,29)(21,85,87,126,22,74,83,119,47,58,82,118,42,106,107,124)(26,53,90,114,79,52,92,117,91,69,109,123,86,72,102,113);(1,3)(2,11)(4,26)(5,34)(6,42)(7,47)(8,23)(9,49)(10,24)(12,58)(13,63)(14,69)(15,72)(16,54)(17,56)(18,60)(19,30)(20,39)(21,45)(22,27)(25,76)(28,92)(29,80)(31,86)(32,79)(33,91)(35,102)(36,87)(37,83)(38,84)(40,100)(41,52)(43,108)(44,82)(46,53)(48,55)(50,103)(51,66)(57,78)(59,119)(61,74)(62,85)(64,117)(65,114)(67,115)(68,107)(70,75)(71,113)(73,88)(77,98)(81,97)(89,106)(90,95)(93,110)(94,111)(96,109)(99,120)(101,112)(104,116)(105,123)(118,127)(121,126)(122,124)(125,128)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,h^b=hi,d^c=dghi,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1007);