Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12137 |
| commutator, Frattini, cosocle | #408 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(1,2,9,16)(3,17,23,67)(4,24,29,59)(5,30,36,56)(6,13,41,62)(7,15,8,14)(10,43,48,19)(11,38,52,26)(12,35,57,34)(18,68,73,123)(20,65,79,114)(21,66,22,53)(25,84,88,121)(27,91,28,83)(31,47,76,46)(32,97,87,82)(33,94,39,106)(37,102,80,120)(40,63,42,61)(44,107,112,127)(45,78,115,75)(49,98,93,99)(50,90,51,92)(54,86,118,105)(55,101,60,100)(58,96,116,104)(64,77,117,70)(69,81,95,122)(71,126,72,124)(74,109,85,113)(89,108,103,119)(110,125,111,128);(1,3,9,23)(2,10,16,48)(4,18,29,73)(5,31,36,76)(6,37,41,80)(7,22,8,21)(11,44,52,112)(12,53,57,66)(13,58,62,116)(14,47,15,46)(17,56,67,30)(19,35,43,34)(20,42,79,40)(24,81,59,122)(25,85,88,74)(26,89,38,103)(27,72,28,71)(32,95,87,69)(33,78,39,75)(45,63,115,61)(49,117,93,64)(50,111,51,110)(54,119,118,108)(55,114,60,65)(68,121,123,84)(70,90,77,92)(82,124,97,126)(83,113,91,109)(86,125,105,128)(94,120,106,102)(96,101,104,100)(98,107,99,127);(1,4,8,27,9,29,7,28)(2,11,15,50,16,52,14,51)(3,18,22,71,23,73,21,72)(5,32,35,25,36,87,34,88)(6,38,42,92,41,26,40,90)(10,44,47,110,48,112,46,111)(12,54,56,49,57,118,30,93)(13,59,63,91,62,24,61,83)(17,64,66,108,67,117,53,119)(19,74,76,69,43,85,31,95)(20,77,80,89,79,70,37,103)(33,99,101,105,39,98,100,86)(45,113,116,81,115,109,58,122)(55,84,106,97,60,121,94,82)(65,123,102,126,114,68,120,124)(75,107,104,128,78,127,96,125)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=fi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fi,g^a=gh,c^b=ch,d^b=df,e^b=eghi,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58435);