Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#7547 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C22 | #1423 |
| prime radical | C23 | Q64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
110 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);(3,4,7,14,11,17,10,40)(5,18,21,55,25,58,24,76)(6,26,28,77,31,80,30,86)(8,34,32,87,37,89,36,90)(9,38,33,53,39,92,35,16)(12,41,42,67,45,66,93,20)(13,46,47,63,49,62,94,19)(15,51,50,82,52,81,88,27)(22,70,68,119,73,121,72,122)(23,74,69,109,75,123,71,57)(29,84,83,126,85,127,60,79)(43,99,98,129,100,130,105,111)(44,65,91,97,101,118,128,110)(48,61,78,102,103,115,125,114)(54,96,104,113,106,112,124,59)(56,95,107,117,108,116,120,64);(3,5,19,59,8,22,60,110,35,71,114,130,81,116,30,66,10,24,62,112,36,72,85,118,39,75,115,129,82,117,31,67,11,25,63,113,37,73,83,97,33,69,102,99,51,95,28,41,7,21,46,96,32,68,29,65,9,23,61,111,27,64,6,20)(4,12,26,56,15,43,78,74,38,91,84,119,87,104,47,55,14,42,77,107,50,98,103,109,53,101,126,121,89,106,49,58,17,45,80,108,52,100,125,123,92,128,127,122,90,124,94,76,40,93,86,120,88,105,48,57,16,44,79,70,34,54,13,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60771);