Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | D4 |
| commutator | C4 | C4×C32 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C22 | C22⋊C32 |
| prime radical | C23 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 132 points:
(5,6,9,16,10,17,32,51,11,18,33,52,36,55,79,100,12,19,34,53,37,56,80,101,39,58,82,103,84,105,120,128,13,20,35,54,38,57,81,102,40,59,83,104,85,106,121,129,8,15,28,47,29,48,73,95,30,49,74,96,75,97,116,126)(7,21,23,60,24,61,66,107,25,62,67,108,68,109,113,130,22,45,63,50,64,91,110,98,65,92,111,99,112,124,131,127,26,46,31,76,69,93,77,117,70,94,78,118,114,125,119,132,27,14,41,42,71,43,86,88,72,44,87,89,115,90,122,123);(1,2,3,4)(5,7)(6,14)(8,27)(9,31)(10,24)(11,25)(12,22)(13,26)(15,46)(16,50)(17,43)(18,44)(19,21)(20,45)(23,35)(28,63)(29,71)(30,72)(32,77)(33,78)(34,41)(36,68)(37,64)(38,69)(39,65)(40,70)(42,54)(47,60)(48,93)(49,94)(51,98)(52,99)(53,76)(55,90)(56,61)(57,91)(58,62)(59,92)(66,81)(67,83)(73,110)(74,111)(75,115)(79,119)(80,86)(82,87)(84,112)(85,114)(88,102)(89,104)(95,107)(96,108)(97,125)(100,127)(101,117)(103,118)(105,109)(106,124)(113,121)(116,131)(120,122)(123,129)(126,130)(128,132)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^b=ci,d^b=di224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1956);