Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151015 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11982 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,6,14,8,44,36,13,7,42,39,66,43,59,38,104,10,17,41,63,46,16,37,61,9,15,40,60,35,105,5,33)(3,18,21,64,23,86,78,65,22,84,81,51,85,113,80,50,25,68,83,49,87,57,79,106,24,62,82,11,45,48,20,76)(4,26,28,90,30,100,97,103,29,12,52,54,99,107,27,58,32,92,96,116,102,91,95,89,31,56,98,117,94,55,34,53)(19,69,71,47,73,109,124,128,72,67,120,88,126,115,70,118,75,123,93,112,127,122,114,111,74,108,125,110,101,119,77,121);(1,3,10,25)(2,11,17,51)(4,19,32,75)(5,21,38,83)(6,20,41,80)(7,24,9,22)(8,45,46,85)(12,47,56,112)(13,57,61,86)(14,62,63,84)(15,64,42,49)(16,65,44,106)(18,66,68,60)(23,43,87,35)(26,88,92,110)(27,93,34,71)(28,77,96,70)(29,74,31,72)(30,101,102,126)(33,48,104,113)(36,82,37,81)(39,79,40,78)(50,105,76,59)(52,114,98,124)(53,115,58,119)(54,69,117,123)(55,118,107,121)(67,116,108,90)(73,99,127,94)(89,122,103,109)(91,111,100,128)(95,120,97,125);(1,4)(2,12,17,56)(3,19)(5,34)(6,28)(7,29)(8,30)(9,31)(10,32)(11,47,51,112)(13,58,61,53)(14,54,63,117)(15,26,42,92)(16,55,44,107)(18,67,68,108)(20,77)(21,71)(22,72)(23,73)(24,74)(25,75)(27,38)(33,103,104,89)(35,94)(36,97)(37,95)(39,52)(40,98)(41,96)(43,99)(45,101)(46,102)(48,109,113,122)(49,110,64,88)(50,111,76,128)(57,119,86,115)(59,91,105,100)(60,90,66,116)(62,69,84,123)(65,118,106,121)(70,80)(78,124)(79,114)(81,120)(82,125)(83,93)(85,126)(87,127)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fh,d^2=gh,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58212);