Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94843 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7546 |
| commutator | C64 | C2×C4 |
| Frattini, cosocle | C2×C64 | C22 |
| prime radical | C22 | SD128 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(2,9,11,33)(3,16)(4,22)(5,23)(6,27)(7,30)(10,38)(12,49,45,83)(13,54,46,108)(14,57,47,110)(15,58,48,37)(17,26)(18,66)(19,42)(20,44)(21,43)(24,77)(25,65)(28,81)(29,76)(31,80)(32,82,51,34)(35,90,87,50)(36,92,88,55)(39,95,96,86)(40,100)(41,103)(52,115,106,128)(53,116,107,98)(56,118,109,97)(59,60)(61,67)(62,102)(63,64)(68,122)(69,120)(70,105)(71,72)(73,78)(74,75)(79,121)(84,127,114,111)(85,126,112,99)(89,119,125,123)(91,113,117,124)(93,101)(94,104);(1,2,10,39,4,12,40,97,23,50,101,115,72,112,67,57,6,14,42,99,24,52,18,35,74,109,60,49,65,86,3,9,30,48,16,34,78,107,59,54,81,124,66,111,121,119,19,36,80,92,61,89,63,84,93,117,29,46,100,116,17,32,38,58,8,11,44,96,26,45,103,118,76,90,122,128,64,85,94,110,31,47,105,126,79,106,69,87,28,56,62,83,73,95,21,33,7,15,43,82,25,53,102,108,75,113,120,127,77,123,70,88,27,55,104,125,71,114,68,91,5,13,41,98,22,51,20,37)Polycyclic presentation:
2^9|a^2=d,b^2=ceh,c^2=efi,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,d^b=di86 conjugacy classes — view the full character table.
| maximal subgroup of | #72230770 |
| maximal normal subgroup of | #72230770 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2039);