Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145499 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10513 |
| commutator, Frattini, cosocle | #405 | C23 |
| prime radical | C23 | #472 |
| LCS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,8,14)(3,15,20,61)(4,21,26,77)(5,27,33,58)(6,12,35,55)(7,37,30,13)(9,39,43,17)(10,44,48,24)(11,49,53,32)(16,62,66,124)(18,60,70,122)(19,72,38,50)(22,78,81,123)(23,82,29,91)(25,54,63,76)(28,57,90,42)(31,74,96,121)(34,47,86,101)(36,99,94,56)(40,103,106,125)(41,102,108,68)(45,111,113,126)(46,98,51,95)(52,110,118,85)(59,97,104,65)(64,119,67,105)(69,115,80,112)(71,127,100,116)(73,88,107,79)(75,93,117,84)(83,114,92,87)(89,120,128,109);(1,3,8,20)(2,9,14,43)(4,22,26,81)(5,28,33,90)(6,18,35,70)(7,38,30,19)(10,45,48,113)(11,50,53,72)(12,41,55,108)(13,57,37,42)(15,58,61,27)(16,63,66,25)(17,49,39,32)(21,73,77,107)(23,67,29,64)(24,80,44,69)(31,89,96,128)(34,97,86,65)(36,100,94,71)(40,101,106,47)(46,59,51,104)(52,116,118,127)(54,119,76,105)(56,120,99,109)(60,121,122,74)(62,117,124,75)(68,110,102,85)(78,91,123,82)(79,93,88,84)(83,125,92,103)(87,115,114,112)(95,126,98,111);(1,4,5,29,30,63,49,84,8,26,33,23,7,25,32,93)(2,10,11,51,37,101,58,114,14,48,53,46,13,47,27,87)(3,16,17,67,38,81,90,79,20,66,39,64,19,22,28,88)(6,34,31,95,94,44,110,83,35,86,96,98,36,24,85,92)(9,40,15,59,57,113,72,112,43,106,61,104,42,45,50,115)(12,54,52,82,99,77,121,75,55,76,118,91,56,21,74,117)(18,69,68,126,100,97,128,103,70,80,102,111,71,65,89,125)(41,107,60,78,120,119,127,124,108,73,122,123,109,105,116,62)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=dh,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cf,d^b=dg,f^b=fh,g^b=gi,d^c=di,e^c=egh,f^c=fgh,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52697);