Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501080 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24378 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,7,13,35,56,87,76,8,14,39,60,6,12,34,55)(3,15,19,63,18,45,73,57,21,44,36,96,20,9,40,43)(4,22,26,54,33,94,95,49,27,78,89,48,25,58,37,10)(5,28,30,91,29,53,92,59,32,52,38,99,31,11,50,51)(16,64,66,41,23,79,81,46,68,108,97,102,67,112,74,61)(17,42,70,104,71,62,100,115,72,105,75,106,69,103,123,116)(24,82,84,47,83,110,98,109,86,113,88,111,85,77,93,90)(65,118,120,107,80,117,125,127,121,126,122,114,119,128,124,101);(1,3)(2,9)(4,23)(5,17)(6,20)(7,36)(8,21)(10,46)(11,42)(12,44)(13,57)(14,45)(15,56)(16,25)(18,35)(19,39)(22,64)(24,80)(26,74)(27,67)(28,62)(29,71)(30,75)(31,69)(32,72)(33,68)(34,73)(37,97)(38,70)(40,87)(41,48)(43,60)(47,107)(49,61)(50,100)(51,106)(52,103)(53,105)(54,102)(55,63)(58,112)(59,104)(65,85)(66,95)(76,96)(77,117)(78,108)(79,94)(81,89)(82,126)(83,121)(84,124)(86,119)(88,125)(90,101)(91,116)(92,123)(93,122)(98,120)(99,115)(109,127)(110,128)(111,114)(113,118);(1,4,8,27)(2,10,14,49)(3,16,21,68)(5,24,32,86)(6,33,35,25)(7,37,39,95)(9,41,45,102)(11,47,53,111)(12,54,56,48)(13,58,60,94)(15,61,44,46)(17,65,72,121)(18,67,20,23)(19,74,36,81)(22,76,78,55)(26,87,89,34)(28,90,52,109)(29,85,31,83)(30,93,38,98)(40,66,73,97)(42,101,105,127)(43,64,57,108)(50,84,92,88)(51,82,59,113)(62,114,103,107)(63,112,96,79)(69,80,71,119)(70,124,75,125)(77,99,110,91)(100,122,123,120)(104,128,106,117)(115,126,116,118);(1,5)(2,11)(3,17)(4,24)(6,31)(7,38)(8,32)(9,42)(10,47)(12,52)(13,59)(14,53)(15,62)(16,65)(18,71)(19,75)(20,69)(21,72)(22,77)(23,80)(25,85)(26,88)(27,86)(28,56)(29,35)(30,39)(33,83)(34,92)(36,70)(37,98)(40,100)(41,101)(43,106)(44,103)(45,105)(46,107)(48,90)(49,111)(50,87)(51,60)(54,109)(55,91)(57,104)(58,113)(61,114)(63,116)(64,117)(66,122)(67,119)(68,121)(73,123)(74,125)(76,99)(78,110)(79,126)(81,124)(82,94)(84,89)(93,95)(96,115)(97,120)(102,127)(108,128)(112,118)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408277);