Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×#2184 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
737 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)(4,23)(5,11)(6,32)(7,36)(8,15)(9,19)(10,26)(12,34)(13,31)(14,30)(17,59)(18,56)(20,69)(21,57)(22,58)(24,47)(25,76)(27,45)(28,46)(29,84)(33,53)(35,91)(37,50)(38,49)(39,96)(40,65)(41,67)(42,92)(43,68)(44,102)(48,86)(51,110)(52,94)(54,88)(55,62)(60,111)(61,71)(63,118)(64,112)(66,109)(70,108)(72,79)(73,75)(74,120)(77,113)(78,89)(80,123)(81,95)(82,114)(83,107)(85,101)(87,90)(93,100)(97,125)(98,115)(99,117)(103,119)(104,126)(105,122)(106,124)(116,128)(121,127);(1,3)(2,9)(4,17)(5,18)(6,33)(7,37)(8,22)(10,39)(11,40)(12,50)(13,41)(14,53)(15,43)(16,32)(19,34)(20,30)(21,31)(23,71)(24,60)(25,77)(26,62)(27,82)(28,64)(29,85)(35,93)(36,42)(38,66)(44,103)(45,106)(46,99)(47,98)(48,69)(49,101)(51,108)(52,109)(54,100)(55,89)(56,84)(57,90)(58,91)(59,76)(61,79)(63,75)(65,86)(67,110)(68,94)(70,88)(72,114)(73,113)(74,121)(78,124)(80,125)(81,119)(83,116)(87,92)(95,97)(96,102)(104,128)(105,118)(107,127)(111,120)(112,123)(115,122)(117,126);(1,4)(2,10)(3,17)(5,24)(6,25)(7,27)(8,28)(9,39)(11,23)(12,44)(13,46)(14,45)(15,47)(16,55)(18,60)(19,61)(20,63)(21,62)(22,64)(26,31)(29,74)(30,75)(32,89)(33,77)(34,79)(35,80)(36,95)(37,82)(38,83)(40,71)(41,99)(42,97)(43,98)(48,72)(49,73)(50,103)(51,104)(52,105)(53,106)(54,107)(56,59)(57,112)(58,111)(65,115)(66,116)(67,96)(68,117)(69,114)(70,119)(76,84)(78,87)(81,88)(85,121)(86,122)(90,123)(91,120)(92,124)(93,125)(94,126)(100,127)(101,113)(102,110)(108,128)(109,118);(1,5,8,31)(2,11,15,13)(3,18,22,21)(4,24,28,26)(6,29,35,87)(7,30,38,88)(9,40,43,41)(10,23,47,46)(12,48,52,51)(14,49,54,36)(16,56,58,57)(17,60,64,62)(19,65,68,67)(20,66,70,37)(25,74,80,78)(27,75,83,81)(32,84,91,90)(33,85,93,92)(34,86,94,110)(39,71,98,99)(42,53,101,100)(44,72,105,104)(45,73,107,95)(50,69,109,108)(55,59,111,112)(61,115,117,96)(63,116,119,82)(76,120,123,89)(77,121,125,124)(79,122,126,102)(97,106,113,127)(103,114,118,128)Polycyclic presentation:
2^9|d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420044);