Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500753 |
| 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 | |
990 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)(3,17)(4,25)(5,32)(6,38)(7,42)(8,14)(9,16)(10,20)(11,29)(12,34)(13,41)(15,40)(18,70)(19,76)(21,49)(22,50)(23,52)(24,51)(26,47)(27,88)(28,56)(30,62)(31,65)(33,84)(35,96)(36,66)(37,97)(39,63)(43,102)(44,69)(45,60)(46,64)(48,82)(53,75)(54,94)(55,95)(57,93)(58,90)(59,101)(61,89)(67,73)(68,78)(71,123)(72,112)(74,106)(77,114)(79,113)(80,109)(81,111)(83,120)(85,127)(86,105)(87,107)(91,115)(92,116)(98,104)(99,118)(100,119)(103,110)(108,125)(117,126)(121,128)(122,124);(1,3,6,20,9,24,41,23)(2,10,13,17,16,52,38,51)(4,26,28,73,31,87,39,86)(5,19,34,78,37,82,89,81)(7,43,8,44,40,22,46,21)(11,53,55,112,57,70,62,106)(12,48,32,68,61,76,97,111)(14,50,42,69,64,102,15,49)(18,29,72,30,75,93,74,95)(25,67,63,47,65,105,56,107)(27,85,33,98,92,125,99,128)(35,80,45,103,101,83,36,79)(54,114,58,117,115,122,119,123)(59,113,60,120,96,110,66,109)(71,90,124,94,126,100,77,91)(84,108,88,104,118,127,116,121);(1,4,9,31)(2,11,16,57)(3,18,24,75)(5,33,37,99)(6,39,41,28)(7,30,40,95)(8,29,46,93)(10,47,52,107)(12,58,61,119)(13,62,38,55)(14,63,64,56)(15,65,42,25)(17,67,51,105)(19,77,82,124)(20,74,23,72)(21,26,44,87)(22,73,43,86)(27,89,92,34)(32,54,97,115)(35,100,101,90)(36,91,45,94)(48,108,76,127)(49,70,69,53)(50,112,102,106)(59,118,96,84)(60,88,66,116)(68,121,111,104)(71,78,126,81)(79,128,103,98)(80,125,83,85)(109,117,120,123)(110,122,113,114);(1,5,6,34,9,37,41,89)(2,12,13,32,16,61,38,97)(3,19,20,78,24,82,23,81)(4,27,28,33,31,92,39,99)(7,35,8,45,40,101,46,36)(10,48,17,68,52,76,51,111)(11,54,55,58,57,115,62,119)(14,59,42,60,64,96,15,66)(18,71,72,124,75,126,74,77)(21,79,43,80,44,103,22,83)(25,84,63,88,65,118,56,116)(26,85,73,98,87,125,86,128)(29,90,30,94,93,100,95,91)(47,104,105,127,107,121,67,108)(49,109,50,113,69,120,102,110)(53,114,112,117,70,122,106,123)Polycyclic presentation:
2^9|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=gh,h^a=hi,c^b=cg,f^b=fhi,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, 407951);