Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1415 |
| commutator | C32 | C42 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C2 | #7471 |
| prime radical | C23 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,13,10,17,9,46)(3,18,20,63,25,53,24,83)(4,26,29,59,33,88,32,14)(6,37,34,55,40,99,30,12)(7,41,35,98,38,100,43,103)(8,44,36,61,45,105,42,16)(11,47,48,27,52,69,106,19)(15,60,56,92,57,91,85,28)(21,72,67,62,77,117,76,116)(22,78,70,112,73,123,80,122)(23,81,71,111,82,126,79,65)(31,95,93,128,96,58,94,87)(39,101,97,127,102,54,90,86)(49,89,84,120,113,66,64,107)(50,114,109,125,110,75,118,119)(51,68,104,108,115,74,124,121);(1,3,6,21,38,73,96,68,42,79,90,125,60,107,33,19,9,24,30,76,35,70,93,121,45,82,102,114,28,66,29,69,10,25,40,77,7,22,31,74,36,71,97,119,91,120,4,27,5,20,34,67,43,80,94,108,8,23,39,75,92,89,32,47)(2,11,14,49,57,110,101,65,44,104,87,123,103,72,55,18,13,48,26,84,85,118,127,81,61,115,95,122,41,62,99,63,17,52,59,113,15,50,54,111,105,124,128,78,98,117,12,53,46,106,88,64,56,109,86,126,16,51,58,112,100,116,37,83)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=i,e^2=fi,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321140, #72321181, #72321207, … |
| maximal normal subgroup of | #72321140, #72321181, #72321207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1966);