Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500749 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24248 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61108 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,16)(3,17,24,49)(4,25,31,89)(5,32,37,99)(6,38,41,13)(7,15,40,66)(8,14,44,63)(10,23,50,20)(11,51,56,29)(12,57,61,34)(18,70,75,52)(19,76,82,46)(21,47,83,69)(22,64,42,48)(26,90,84,45)(27,92,95,115)(28,96,39,55)(30,62,98,54)(33,88,101,119)(35,102,104,59)(36,60,43,65)(53,94,116,97)(58,103,118,93)(67,85,87,73)(68,81,111,78)(71,123,127,117)(72,112,74,107)(77,114,124,122)(79,113,105,110)(80,120,86,109)(91,108,126,128)(100,121,125,106);(1,3,6,20,9,24,41,23)(2,10,13,17,16,50,38,49)(4,26,28,85,31,84,39,73)(5,19,34,78,37,82,57,81)(7,42,8,21,40,22,44,83)(11,52,54,112,56,70,62,107)(12,46,32,68,61,76,99,111)(14,64,15,47,63,48,66,69)(18,29,72,30,75,51,74,98)(25,87,55,90,89,67,96,45)(27,91,33,100,95,126,101,125)(35,80,43,79,104,86,36,105)(53,114,58,117,116,122,118,123)(59,110,65,109,102,113,60,120)(71,93,124,97,127,103,77,94)(88,128,92,121,119,108,115,106);(1,4,9,31)(2,11,16,56)(3,18,24,75)(5,33,37,101)(6,39,41,28)(7,30,40,98)(8,29,44,51)(10,45,50,90)(12,58,61,118)(13,62,38,54)(14,55,63,96)(15,25,66,89)(17,67,49,87)(19,77,82,124)(20,74,23,72)(21,84,83,26)(22,85,42,73)(27,57,95,34)(32,53,99,116)(35,103,104,93)(36,94,43,97)(46,108,76,128)(47,70,69,52)(48,112,64,107)(59,119,102,88)(60,115,65,92)(68,121,111,106)(71,78,127,81)(79,100,105,125)(80,126,86,91)(109,117,120,123)(110,122,113,114);(1,5,6,34,9,37,41,57)(2,12,13,32,16,61,38,99)(3,19,20,78,24,82,23,81)(4,27,28,33,31,95,39,101)(7,35,8,43,40,104,44,36)(10,46,17,68,50,76,49,111)(11,53,54,58,56,116,62,118)(14,59,15,65,63,102,66,60)(18,71,72,124,75,127,74,77)(21,79,22,86,83,105,42,80)(25,88,55,92,89,119,96,115)(26,91,85,100,84,126,73,125)(29,93,30,97,51,103,98,94)(45,106,87,128,90,121,67,108)(47,109,48,113,69,120,64,110)(52,114,112,117,70,122,107,123)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=i,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407948);