Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143173 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10026 |
| commutator, Frattini, cosocle | #405 | C23 |
| prime radical | #575 | D4 |
| LCS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,10,18)(3,19,25,73)(4,26,33,92)(5,34,41,64)(6,15,43,66)(7,14,42,67)(8,17,9,16)(11,47,52,21)(12,53,59,29)(13,40,65,39)(20,74,78,122)(22,81,23,84)(24,87,46,60)(27,93,76,71)(28,95,100,124)(30,57,101,58)(31,56,32,91)(35,70,106,51)(36,107,99,90)(37,62,111,104)(38,69,45,63)(44,112,83,72)(48,96,117,128)(49,109,50,105)(54,79,115,113)(55,102,103,98)(61,97,123,110)(68,80,119,86)(75,118,127,120)(77,125,88,121)(82,114,85,126)(89,108,116,94);(1,3)(2,11)(4,27)(5,35)(6,23)(7,44)(8,46)(9,24)(10,25)(12,54)(13,60)(14,50)(15,68)(16,70)(17,51)(18,52)(19,34)(20,30)(21,39)(22,43)(26,89)(28,96)(29,88)(31,82)(32,85)(33,76)(36,108)(37,109)(38,86)(40,47)(41,106)(42,83)(45,80)(48,56)(49,67)(53,77)(55,122)(57,118)(58,120)(59,115)(61,125)(62,84)(63,112)(64,73)(65,87)(66,119)(69,72)(71,124)(74,103)(75,97)(78,101)(79,102)(81,104)(90,114)(91,117)(92,116)(93,95)(94,99)(98,113)(100,128)(105,111)(107,126)(110,127)(121,123);(1,4,10,33)(2,12,18,59)(3,20,25,78)(5,36,41,99)(6,30,43,101)(7,29,42,53)(8,32,9,31)(11,48,52,117)(13,61,65,123)(14,56,67,91)(15,26,66,92)(16,58,17,57)(19,71,73,93)(21,79,47,113)(22,82,23,85)(24,88,46,77)(27,83,76,44)(28,39,100,40)(34,103,64,55)(35,75,106,127)(37,110,111,97)(38,98,45,102)(49,118,50,120)(51,89,70,116)(54,119,115,68)(60,114,87,126)(62,107,104,90)(63,95,69,124)(72,121,112,125)(74,84,122,81)(80,94,86,108)(96,105,128,109)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fg,g^c=gi,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50371);