Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153083 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2215 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #508 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 132 points:
(1,2)(3,4)(5,6,13,20)(7,21,27,76)(8,28,34,67)(9,35,40,69)(10,41,46,64)(11,47,44,109)(12,50,49,19)(14,52,57,23)(15,43,62,31)(16,45,65,38)(17,39,70,107)(18,71,68,37)(22,77,81,114)(24,55,86,117)(25,54,84,119)(26,66,51,75)(29,74,83,121)(30,95,101,111)(32,103,102,124)(33,105,98,90)(36,72,85,127)(42,56,82,73)(48,63,110,118)(53,80,115,91)(58,93,116,78)(59,97,89,128)(60,100,88,132)(61,99,87,108)(79,113,92,120)(94,125,106,130)(96,129,112,123)(104,122,131,126);(5,7,13,27)(6,14,20,57)(8,29,34,83)(9,36,40,85)(10,42,46,82)(11,48,44,110)(12,51,49,26)(15,58,62,116)(16,63,65,118)(17,66,70,75)(18,72,68,127)(19,73,50,56)(21,69,76,35)(22,43,81,31)(23,45,52,38)(24,39,86,107)(25,71,84,37)(28,53,67,115)(30,96,101,112)(32,104,102,131)(33,106,98,94)(41,54,64,119)(47,55,109,117)(59,123,89,129)(60,125,88,130)(61,126,87,122)(74,111,121,95)(77,105,114,90)(78,97,93,128)(79,100,92,132)(80,99,91,108)(103,113,124,120);(2,4)(5,8,9,30,37,97,10,43,12,33,39,100,44,102,45,108,13,34,40,101,71,128,46,31,49,98,107,132,11,32,38,99)(6,15,16,59,47,111,17,67,19,61,64,124,68,88,69,105,20,62,65,89,109,95,70,28,50,87,41,103,18,60,35,90)(7,22,23,78,48,112,24,83,26,80,82,131,84,92,85,106,27,81,52,93,110,96,86,29,51,91,42,104,25,79,36,94)(14,53,21,74,72,129,54,116,56,114,75,130,117,120,118,126,57,115,76,121,127,123,119,58,73,77,66,125,55,113,63,122)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=d,d^2=gh,e^2=gi,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fghi,g^b=gh,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60512);