Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151185 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12102 |
| commutator, Frattini, cosocle | C2×M32 | C23 |
| prime radical | #13312 | C2 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
275 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,15)(4,22)(5,28)(6,36)(7,38)(8,14)(9,17)(10,24)(11,33)(12,37)(13,31)(16,61)(18,59)(19,73)(20,60)(21,58)(23,79)(25,46)(26,47)(27,48)(29,54)(30,94)(32,90)(34,56)(35,89)(39,49)(40,99)(41,101)(42,71)(43,92)(44,69)(45,106)(50,96)(51,97)(52,66)(53,68)(55,74)(57,63)(62,76)(64,121)(65,117)(67,102)(70,116)(72,113)(75,112)(77,83)(78,81)(80,118)(82,115)(84,88)(85,100)(86,119)(87,111)(91,103)(93,114)(95,108)(98,105)(104,120)(107,122)(109,126)(110,125)(123,128)(124,127);(1,3,8,20)(2,9,14,44)(4,16,26,65)(5,29,34,93)(6,18,37,21)(7,39,40,72)(10,41,48,104)(11,49,52,113)(12,53,36,42)(13,54,55,114)(15,56,60,28)(17,66,69,33)(19,74,75,31)(22,76,47,102)(23,80,30,91)(24,63,27,70)(25,86,87,123)(32,92,35,98)(38,43,99,105)(45,107,109,121)(46,110,111,127)(50,73,51,112)(57,115,116,88)(58,89,59,90)(61,94,117,79)(62,95,67,83)(64,85,122,81)(68,96,71,97)(77,119,108,128)(78,118,100,103)(82,124,84,125)(101,126,120,106);(1,4)(2,10)(3,16)(5,30)(6,27)(7,25)(8,26)(9,41)(11,45)(12,47)(13,46)(14,48)(15,57)(17,67)(18,70)(19,64)(20,65)(21,63)(22,36)(23,34)(24,37)(28,88)(29,91)(31,81)(32,82)(33,95)(35,84)(38,100)(39,86)(40,87)(42,76)(43,103)(44,104)(49,107)(50,77)(51,108)(52,109)(53,102)(54,110)(55,111)(56,115)(58,117)(59,61)(60,116)(62,69)(66,83)(68,120)(71,101)(72,123)(73,119)(74,85)(75,122)(78,99)(79,89)(80,93)(90,94)(92,124)(96,106)(97,126)(98,125)(105,118)(112,128)(113,121)(114,127)Polycyclic presentation:
2^9|b^2=i,d^2=fh,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=df,e^b=egi,f^b=fh,h^b=hi,d^c=dg,e^c=ei,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58382);