Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #506304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28165 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
536 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)(3,15)(4,21)(5,11)(6,30)(7,36)(8,13)(9,18)(10,23)(12,34)(14,33)(16,56)(17,39)(19,64)(20,41)(22,70)(24,79)(25,71)(26,80)(27,84)(28,49)(29,90)(31,92)(32,53)(35,69)(37,51)(38,76)(40,95)(42,59)(43,73)(44,78)(45,96)(46,82)(47,89)(48,87)(50,66)(52,77)(54,61)(55,63)(57,114)(58,68)(60,100)(62,106)(65,98)(67,111)(72,122)(74,121)(75,125)(81,120)(83,99)(85,112)(86,110)(88,119)(91,108)(93,101)(94,107)(97,116)(102,126)(103,118)(104,128)(105,124)(109,117)(113,115)(123,127);(1,3)(2,9,13,15)(4,16)(5,26,28,82)(6,31)(7,20)(8,18)(10,42,44,38)(11,46)(12,50,52,40)(14,41,36,55)(17,29,61,87)(19,35)(21,68,71,56)(22,72,74,81)(23,76)(24,60)(25,58)(27,85,88,83)(30,64,69,92)(32,93)(33,63)(34,95)(37,67)(39,90)(43,103)(45,101,53,98)(47,107)(48,54)(49,80)(51,100,79,111)(57,75,115,91)(59,78)(62,117,94,89)(65,96)(66,77)(70,120)(73,123,124,118)(84,99)(86,116,128,126)(97,104)(102,110)(105,127)(106,109)(108,113)(112,119)(114,125)(121,122);(1,4,8,25)(2,10,13,44)(3,16,18,58)(5,22,28,74)(6,32,35,96)(7,37,33,24)(9,38,15,42)(11,43,49,105)(12,51,52,79)(14,53,36,45)(17,57,61,115)(19,65,31,93)(20,67,63,60)(21,69,71,30)(23,77,78,34)(26,81,82,72)(27,86,88,128)(29,91,87,75)(39,97,54,102)(40,100,50,111)(41,101,55,98)(46,103,80,127)(47,108,109,125)(48,110,90,104)(56,92,68,64)(59,95,76,66)(62,118,94,123)(70,119,121,84)(73,117,124,89)(83,116,85,126)(99,120,112,122)(106,114,107,113);(1,5,8,28)(2,11,13,49)(3,17)(4,22,25,74)(6,27,35,88)(7,29,33,87)(9,39)(10,43,44,105)(12,47,52,109)(14,48,36,90)(15,54)(16,57)(18,61)(19,62)(20,26)(21,70,71,121)(23,73,78,124)(24,75,37,91)(30,84,69,119)(31,94)(32,86,96,128)(34,89,77,117)(38,97)(40,99)(41,46)(42,102)(45,104,53,110)(50,112)(51,108,79,125)(55,80)(56,113)(58,115)(59,116)(60,72)(63,82)(64,107)(65,118)(66,83)(67,81)(68,114)(76,126)(85,95)(92,106)(93,123)(98,127)(100,120)(101,103)(111,122)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fi,f^c=fh,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, 413503);