Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C128 · Schur multiplier (H2) = C1 |
| modular maximal cyclic | M512 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C128 | C22 |
| commutator, socle | C2 | C2×C128 |
| prime radical | C22 | C128 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C128 ◁ G | |
| LECS | G ▷ C128 ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 256 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256);(2,130)(4,132)(6,134)(8,136)(10,138)(12,140)(14,142)(16,144)(18,146)(20,148)(22,150)(24,152)(26,154)(28,156)(30,158)(32,160)(34,162)(36,164)(38,166)(40,168)(42,170)(44,172)(46,174)(48,176)(50,178)(52,180)(54,182)(56,184)(58,186)(60,188)(62,190)(64,192)(66,194)(68,196)(70,198)(72,200)(74,202)(76,204)(78,206)(80,208)(82,210)(84,212)(86,214)(88,216)(90,218)(92,220)(94,222)(96,224)(98,226)(100,228)(102,230)(104,232)(106,234)(108,236)(110,238)(112,240)(114,242)(116,244)(118,246)(120,248)(122,250)(124,252)(126,254)(128,256)Polycyclic presentation:
2^9|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bi320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2041);