Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497451 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊C4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,7,15,9,14,8,39)(3,16,20,61,21,60,40,55)(4,22,25,50,26,11,38,49)(5,27,30,12,31,54,41,53)(6,32,35,57,37,96,36,13)(10,33,45,73,44,97,46,19)(17,62,65,111,67,58,66,107)(18,68,69,88,71,51,70,59)(23,77,80,47,82,89,81,75)(24,83,86,76,87,95,99,48)(28,90,91,104,92,43,72,102)(29,93,34,56,94,52,98,109)(42,84,101,64,74,116,103,117)(63,105,113,112,114,119,118,110)(78,100,115,127,123,120,126,128)(79,106,124,108,125,121,85,122);(1,3)(2,10)(4,17)(5,28)(6,33)(7,20)(8,40)(9,21)(11,42)(12,51)(13,55)(14,44)(15,45)(16,32)(18,29)(19,36)(22,74)(23,78)(24,84)(25,65)(26,67)(27,88)(30,91)(31,92)(34,69)(35,73)(37,97)(38,66)(39,46)(41,72)(43,52)(47,105)(48,107)(49,101)(50,103)(53,68)(54,59)(56,104)(57,61)(58,95)(60,96)(62,83)(63,79)(64,86)(70,98)(71,94)(75,119)(76,111)(77,110)(80,115)(81,126)(82,123)(85,118)(87,116)(89,112)(90,93)(99,117)(100,106)(102,109)(108,127)(113,124)(114,125)(120,121)(122,128);(1,4,9,26)(2,11,14,22)(3,17,21,67)(5,23,31,82)(6,24,37,87)(7,38,8,25)(10,42,44,74)(12,47,53,75)(13,48,57,76)(15,50,39,49)(16,58,60,62)(18,63,71,114)(19,64,73,117)(20,66,40,65)(27,89,54,77)(28,78,92,123)(29,79,94,125)(30,81,41,80)(32,95,96,83)(33,84,97,116)(34,85,98,124)(35,99,36,86)(43,100,90,120)(45,103,46,101)(51,105,68,119)(52,106,93,121)(55,107,61,111)(56,108,109,122)(59,110,88,112)(69,118,70,113)(72,115,91,126)(102,128,104,127);(1,5)(2,12)(3,18)(4,23)(6,34)(7,30)(8,41)(9,31)(10,43)(11,47)(13,56)(14,53)(15,54)(16,59)(17,63)(19,72)(20,69)(21,71)(22,75)(24,85)(25,80)(26,82)(27,39)(28,33)(29,36)(32,52)(35,94)(37,98)(38,81)(40,70)(42,100)(44,90)(45,102)(46,104)(48,108)(49,89)(50,77)(51,55)(57,109)(58,110)(60,88)(61,68)(62,112)(64,115)(65,113)(66,118)(67,114)(73,91)(74,120)(76,122)(78,84)(79,86)(83,121)(87,124)(92,97)(93,96)(95,106)(99,125)(101,127)(103,128)(105,107)(111,119)(116,123)(117,126)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,h^b=hi,g^c=gi,e^d=eh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404652);