Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C2×D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×D4×C16 |
| prime radical | #575 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,14,7,15,31,50,8,13,32,48,34,49,76,95,9,12,33,46,35,47,77,94,5,27,29,71,30,72,75,111)(3,16,19,52,20,53,59,98,18,10,36,39,58,40,79,86,21,41,60,87,61,88,103,120,28,38,73,84,74,85,112,119)(4,11,23,44,24,45,66,93,25,43,67,91,69,92,109,122,26,42,68,89,70,90,110,121,22,62,64,104,65,105,108,125)(17,51,55,96,56,97,100,123,54,37,78,81,99,82,113,116,57,83,101,117,102,118,124,128,63,80,106,114,107,115,126,127);(1,3)(2,10)(4,17)(5,28)(6,19)(7,20)(8,18)(9,21)(11,37)(12,38)(13,41)(14,39)(15,40)(16,27)(22,63)(23,55)(24,56)(25,54)(26,57)(29,73)(30,74)(31,59)(32,36)(33,60)(34,58)(35,61)(42,80)(43,83)(44,81)(45,82)(46,84)(47,85)(48,87)(49,88)(50,86)(51,62)(52,71)(53,72)(64,106)(65,107)(66,100)(67,78)(68,101)(69,99)(70,102)(75,112)(76,79)(77,103)(89,114)(90,115)(91,117)(92,118)(93,116)(94,119)(95,120)(96,104)(97,105)(98,111)(108,126)(109,113)(110,124)(121,127)(122,128)(123,125);(1,4,6,23,7,24,31,66,8,25,32,67,34,69,76,109,9,26,33,68,35,70,77,110,5,22,29,64,30,65,75,108)(2,11,14,44,15,45,50,93,13,43,48,91,49,92,95,122,12,42,46,89,47,90,94,121,27,62,71,104,72,105,111,125)(3,17,19,55,20,56,59,100,18,54,36,78,58,99,79,113,21,57,60,101,61,102,103,124,28,63,73,106,74,107,112,126)(10,37,39,81,40,82,86,116,41,83,87,117,88,118,120,128,38,80,84,114,85,115,119,127,16,51,52,96,53,97,98,123);(2,12)(10,38)(11,42)(13,27)(14,46)(15,47)(16,41)(37,80)(39,84)(40,85)(43,62)(44,89)(45,90)(48,71)(49,72)(50,94)(51,83)(52,87)(53,88)(81,114)(82,115)(86,119)(91,104)(92,105)(93,121)(95,111)(96,117)(97,118)(98,120)(116,127)(122,125)(123,128)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=be,d^a=di,e^a=ei,e^b=ei176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419945);