Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | C2×SD256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D64 |
| commutator, Frattini, cosocle | C64 | C23 |
| prime radical | C2×D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 130 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 130 points:
(1,2);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130);(4,66)(5,129)(6,64)(7,127)(8,62)(9,125)(10,60)(11,123)(12,58)(13,121)(14,56)(15,119)(16,54)(17,117)(18,52)(19,115)(20,50)(21,113)(22,48)(23,111)(24,46)(25,109)(26,44)(27,107)(28,42)(29,105)(30,40)(31,103)(32,38)(33,101)(34,36)(35,99)(37,97)(39,95)(41,93)(43,91)(45,89)(47,87)(49,85)(51,83)(53,81)(55,79)(57,77)(59,75)(61,73)(63,71)(65,69)(68,130)(70,128)(72,126)(74,124)(76,122)(78,120)(80,118)(82,116)(84,114)(86,112)(88,110)(90,108)(92,106)(94,104)(96,102)(98,100)134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60899);