Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2158 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | #12024 |
| prime radical | C23 | #368 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,13,8,14,37,57,9,16,38,58,40,15,7,39)(3,17,21,61,23,62,78,118,24,64,79,119,81,63,22,80)(4,25,28,84,30,53,29,51,31,52,36,85,94,11,47,50)(5,32,33,54,34,96,68,56,35,98,100,55,67,97,18,12)(10,41,43,75,44,102,109,76,46,103,110,120,45,65,60,20)(19,69,72,122,74,112,73,117,26,86,77,123,89,59,88,116)(27,90,91,49,92,105,66,83,93,127,99,114,121,95,70,113)(42,104,106,71,108,124,115,125,48,111,107,87,82,101,126,128);(1,3,18,65,7,22,67,120,40,81,100,103,38,79,35,76,9,24,68,102,37,78,34,75,8,23,33,41,6,21,5,20)(2,10,32,61,13,43,54,62,14,44,96,118,57,109,56,64,16,46,98,119,58,110,55,63,15,45,97,80,39,60,12,17)(4,26,66,104,29,88,121,125,94,74,91,101,28,77,93,71,31,19,70,111,47,73,92,128,30,89,99,124,36,72,27,87)(11,48,95,116,51,106,83,86,25,82,90,122,85,115,114,59,53,42,105,117,50,107,113,69,52,108,127,123,84,126,49,112);(1,4,8,30,9,31,40,94)(2,11,15,52,16,53,14,25)(3,19,23,74,24,26,81,89)(5,27,34,92,35,93,67,121)(6,36,37,47,38,28,7,29)(10,42,45,82,46,48,44,108)(12,49,55,113,56,114,54,83)(13,51,39,84,58,50,57,85)(17,59,63,86,64,112,62,69)(18,66,33,99,68,70,100,91)(20,71,75,125,76,87,120,128)(21,77,78,88,79,72,22,73)(32,95,97,127,98,105,96,90)(41,101,102,104,103,124,65,111)(43,107,60,115,110,106,109,126)(61,117,80,122,119,116,118,123)Polycyclic presentation:
2^9|a^2=e,b^2=df,c^2=h,d^2=fh,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320649, #72322645, #72322647 |
| maximal normal subgroup of | #72320649, #72322645, #72322647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58254);