Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496130 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21697 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
635 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)(3,16)(4,24,28,76)(5,12)(6,34)(7,14)(8,38)(9,15)(10,20)(11,27,45,29)(13,17)(18,60,64,111)(19,53)(21,55)(22,73)(23,56)(25,46,78,96)(26,79,81,44)(30,43,82,97)(31,84)(32,48)(33,50)(35,51)(36,86)(37,49)(39,88,90,108)(40,67)(41,70)(42,71)(47,59)(52,63,103,65)(54,57)(58,75,80,119)(61,104,113,127)(62,74,114,102)(66,101,115,128)(68,105)(69,107)(72,106)(77,120,110,95)(83,98,121,126)(85,99)(87,100)(89,122,124,116)(91,125,123,112)(92,109)(93,118)(94,117);(1,3,17,57,8,22,36,71,9,23,35,70,7,21,6,20)(2,10,34,55,14,41,51,56,15,42,86,73,38,54,13,16)(4,18,58,108,29,65,81,114,28,64,80,88,27,63,26,62)(5,19,59,109,32,68,85,117,33,69,87,118,37,72,31,67)(11,39,75,60,24,74,79,103,45,90,119,111,76,102,44,52)(12,40,84,106,49,93,100,107,50,94,99,105,48,92,47,53)(25,61,83,116,30,66,110,125,78,113,121,122,82,115,77,112)(43,89,98,104,46,91,120,128,97,124,126,127,96,123,95,101);(1,4)(2,11,15,45)(3,18)(5,30,33,82)(6,26)(7,27)(8,29)(9,28)(10,39,42,90)(12,46)(13,44,51,79)(14,24,38,76)(16,52,56,103)(17,58)(19,66,69,115)(20,62)(21,63)(22,65)(23,64)(25,32,78,37)(31,83,85,121)(34,75,86,119)(35,80)(36,81)(40,91)(41,74,54,102)(43,48)(47,98)(49,97)(50,96)(53,104)(55,60,73,111)(57,108)(59,110,87,77)(61,68,113,72)(67,116,117,122)(70,88)(71,114)(84,120)(89,92)(93,124)(94,123)(95,99)(100,126)(101,105)(106,128)(107,127)(109,125,118,112);(1,5,9,33)(2,12,15,50)(3,19,23,69)(4,25)(6,31,36,85)(7,37,8,32)(10,40,42,94)(11,43)(13,47,51,100)(14,49,38,48)(16,53,56,107)(17,59,35,87)(18,61)(20,67,71,117)(21,72,22,68)(24,46)(26,77)(27,82)(28,78)(29,30)(34,84,86,99)(39,89)(41,93,54,92)(44,95)(45,97)(52,101)(55,106,73,105)(57,109,70,118)(58,83)(60,104)(62,112)(63,115)(64,113)(65,66)(74,91)(75,98)(76,96)(79,120)(80,121)(81,110)(88,122)(90,124)(102,123)(103,128)(108,116)(111,127)(114,125)(119,126)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=fh,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403327);