Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501858 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24897 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#578 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
617 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)(3,17,22,66)(4,24,29,62)(5,30,34,94)(6,35,41,60)(7,15,8,14)(10,43,47,20)(11,42,52,28)(12,53,56,31)(13,39,61,40)(18,67,25,82)(19,45,73,106)(21,57,23,65)(26,84,87,113)(27,58,37,51)(32,59,33,93)(36,46,77,48)(38,54,96,55)(44,98,49,88)(50,89,112,99)(63,78,103,70)(64,115,116,71)(68,119,83,126)(69,108,75,79)(72,107,74,105)(76,104,101,117)(80,95,111,85)(81,100,97,90)(86,92,91,110)(102,123,109,121)(114,122,128,125)(118,124,127,120);(1,3)(2,10)(4,25,29,18)(5,19)(6,36)(7,23)(8,21)(9,22)(11,49,52,44)(12,45)(13,57)(14,48)(15,46)(16,47)(17,35)(20,40)(24,79,62,108)(26,83,87,68)(27,88,37,98)(28,70,42,78)(30,64)(31,76)(32,74)(33,72)(34,73)(38,71)(39,43)(41,77)(50,109,112,102)(51,67,58,82)(53,101)(54,107)(55,105)(56,106)(59,104)(60,66)(61,65)(63,97,103,81)(69,90,75,100)(80,126,111,119)(84,127,113,118)(85,125,95,122)(86,121,91,123)(89,124,99,120)(92,128,110,114)(93,117)(94,116)(96,115);(1,4)(2,11)(3,18)(5,26)(6,37)(7,42)(8,28)(9,29)(10,44)(12,50)(13,58)(14,62)(15,24)(16,52)(17,63)(19,68)(20,75)(21,78)(22,25)(23,70)(27,41)(30,92)(31,95)(32,91)(33,86)(34,87)(35,97)(36,88)(38,99)(39,90)(40,100)(43,69)(45,102)(46,108)(47,49)(48,79)(51,61)(53,85)(54,80)(55,111)(56,112)(57,67)(59,84)(60,81)(64,114)(65,82)(66,103)(71,124)(72,123)(73,83)(74,121)(76,125)(77,98)(89,96)(93,113)(94,110)(101,122)(104,118)(105,126)(106,109)(107,119)(115,120)(116,128)(117,127);(1,5,9,34)(2,12,16,56)(3,19,22,73)(4,26,29,87)(6,38,41,96)(7,33,8,32)(10,45,47,106)(11,50,52,112)(13,59,61,93)(14,55,15,54)(17,64,66,116)(18,68,25,83)(20,76,43,101)(21,74,23,72)(24,80,62,111)(27,89,37,99)(28,91,42,86)(30,60,94,35)(31,39,53,40)(36,71,77,115)(44,102,49,109)(46,107,48,105)(51,113,58,84)(57,104,65,117)(63,114,103,128)(67,118,82,127)(69,122,75,125)(70,123,78,121)(79,126,108,119)(81,110,97,92)(85,100,95,90)(88,124,98,120)Polycyclic presentation:
2^9|a^2=i,b^2=fh,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409057);