Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #418 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | #11631 |
| prime radical | #575 | C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,6,13,8,15,42,63,7,16,41,64,5,33,37,93,9,17,43,65,38,12,56,59,36,14,66,62,39,60,96,116)(3,18,21,69,23,49,81,110,24,57,82,114,34,67,94,118,25,70,83,121,20,10,44,46,22,48,80,109,78,47,101,108)(4,26,29,85,31,84,91,126,30,55,90,61,35,58,95,115,32,54,40,97,92,113,99,127,72,11,50,53,28,52,89,112)(19,71,74,122,73,51,100,111,27,86,88,120,87,45,102,103,76,105,79,125,77,104,124,117,75,68,98,119,123,106,128,107);(1,3)(2,10)(4,27)(5,34)(6,21)(7,24)(8,23)(9,25)(11,51)(12,57)(13,46)(14,67)(15,48)(16,47)(17,49)(18,33)(19,72)(20,38)(22,36)(26,45)(28,73)(29,88)(30,76)(31,87)(32,75)(35,77)(37,94)(39,78)(40,98)(41,82)(42,81)(43,83)(44,56)(50,74)(52,86)(53,111)(54,106)(55,104)(58,68)(59,114)(60,70)(61,117)(62,118)(63,109)(64,108)(65,110)(66,80)(69,93)(71,113)(79,90)(84,105)(85,103)(89,100)(91,102)(92,123)(95,124)(96,101)(97,107)(99,128)(112,120)(115,119)(116,121)(122,127)(125,126);(1,4,7,30,9,32,36,72)(2,11,14,54,17,55,16,26)(3,19,22,75,25,76,24,27)(5,35,38,92,39,28,8,31)(6,40,41,50,43,29,66,90)(10,45,47,104,49,106,67,51)(12,58,33,84,15,52,60,113)(13,61,62,85,65,53,64,97)(18,68,57,71,70,86,48,105)(20,77,34,87,23,73,78,123)(21,79,80,88,83,74,82,98)(37,89,56,91,96,95,42,99)(44,100,94,128,81,124,101,102)(46,107,108,111,110,103,118,117)(59,112,93,127,63,115,116,126)(69,120,114,125,121,119,109,122)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=fi,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,c^b=cfi,d^b=df,f^b=fi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57304);