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:
(1,2,6,13,8,15,42,63,7,16,41,64,5,33,37,93,9,17,43,65,38,12,56,59,36,14,66,62,39,60,97,118)(3,18,21,69,23,49,80,110,24,57,81,116,34,67,95,120,25,70,82,123,20,10,44,46,22,48,79,109,77,47,102,108)(4,26,29,85,31,84,91,127,30,55,90,61,35,58,96,117,32,54,40,98,92,115,100,128,88,11,50,53,28,52,89,114)(19,71,73,124,72,113,101,119,74,104,99,122,94,45,103,105,75,106,78,125,76,51,111,112,27,68,87,121,86,83,126,107);(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)(19,30)(20,38)(22,36)(26,83)(28,76)(29,87)(31,86)(32,74)(35,72)(37,95)(39,77)(40,99)(41,81)(42,80)(43,82)(44,56)(45,54)(50,78)(52,68)(53,112)(55,113)(58,104)(59,116)(60,70)(61,119)(62,120)(63,109)(64,108)(65,110)(66,79)(69,93)(71,84)(73,90)(75,88)(85,107)(89,111)(91,126)(92,94)(96,101)(97,102)(98,105)(100,103)(106,115)(114,121)(117,122)(118,123)(124,127)(125,128);(1,4,7,30,9,32,36,88)(2,11,14,54,17,55,16,26)(3,19,22,27,25,75,24,74)(5,35,38,92,39,28,8,31)(6,40,41,50,43,29,66,90)(10,45,47,51,49,83,67,113)(12,58,33,84,15,52,60,115)(13,61,62,85,65,53,64,98)(18,68,57,71,70,104,48,106)(20,76,34,94,23,72,77,86)(21,78,79,99,82,73,81,87)(37,89,56,91,97,96,42,100)(44,101,95,126,80,111,102,103)(46,107,108,119,110,105,120,112)(59,114,93,128,63,117,118,127)(69,122,116,125,123,121,109,124)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=cf,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, 57303);