Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7548 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1424 |
| commutator | C32 | C42 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C2 | #7471 |
| prime radical | C23 | D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2)(3,17,19,58)(4,12)(5,29)(6,34)(7,38)(8,41)(9,16)(10,18,42,62)(11,25)(13,36)(14,35)(15,39)(20,69,63,57)(21,73,66,110)(22,77,67,119)(23,78,68,60)(24,50)(26,83)(27,86)(28,54)(30,47)(31,91)(32,93)(33,49)(37,97)(40,99)(43,61,101,75)(44,105,102,71)(45,107,103,70)(46,108,100,65)(48,109)(51,81)(52,80)(53,84)(55,79)(56,89)(59,113,112,74)(64,117,115,98)(72,111,116,121)(76,126,118,124)(82,88)(85,127)(87,92)(90,95)(94,96)(104,125,123,114)(106,122,120,128);(1,3,5,20,30,66,80,107,35,74,92,122,79,114,27,65,8,23,33,72,90,118,85,115,40,67,81,105,36,75,28,62,9,19,25,63,6,21,31,70,89,113,109,128,96,125,84,108,39,68,82,116,37,76,26,64,7,22,32,71,24,61,4,18)(2,10,12,43,50,102,93,119,38,98,83,124,97,111,88,78,15,46,53,104,94,120,48,59,56,103,91,110,34,69,11,17,16,42,54,101,13,44,51,77,99,117,127,126,95,121,49,60,41,100,86,123,55,106,87,112,14,45,52,73,47,57,29,58)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,e^2=fi,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321140, #72321194, #72321203, … |
| maximal normal subgroup of | #72321140, #72321194, #72321203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1961);