Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #99 |
| commutator | C2×C4 | C2×C32 |
| Frattini, cosocle | C2×M64 | C22 |
| socle | C2 | #7334 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C2×M64 ◁ G | |
| LECS | G ▷ C2×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,6,14,29,48,7,15,30,49,32,52,76,98,8,11,28,43,33,44,74,91,35,45,75,92,79,93,114,122,9,13,31,47,34,50,77,96,36,51,78,97,80,99,115,124,4,22,24,46,25,64,68,94,26,65,69,95,70,108,110,123)(3,16,18,53,19,54,59,100,20,55,60,101,61,102,106,125,17,10,37,38,56,40,81,84,57,41,82,85,103,88,116,119,21,42,27,71,62,89,72,111,63,90,73,112,107,121,113,127,23,39,58,83,66,86,104,117,67,87,105,118,109,120,126,128);(1,3)(2,10)(4,23)(5,27)(6,19)(7,20)(8,17)(9,21)(11,42)(12,38)(13,39)(14,40)(15,41)(16,22)(18,31)(24,37)(25,66)(26,67)(28,58)(29,72)(30,73)(32,61)(33,56)(34,62)(35,57)(36,63)(43,71)(44,89)(45,90)(46,53)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(54,64)(55,65)(59,77)(60,78)(68,81)(69,82)(70,109)(74,104)(75,105)(76,113)(79,103)(80,107)(91,111)(92,112)(93,121)(94,100)(95,101)(96,117)(97,118)(98,119)(99,120)(102,108)(106,115)(110,116)(114,126)(122,127)(123,125)(124,128)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=ci,d^b=dei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1709);