Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2,5,9,6,3,10,8)(4,7,12,15,13,11,16,14)(17,18)(19,29)(20,25)(21,37)(22,42)(23,26)(24,31)(27,40)(28,39)(30,54)(32,61)(33,55)(34,49)(35,64)(36,67)(38,52)(41,53)(43,51)(44,57)(45,60)(46,69)(47,66)(48,63)(50,59)(56,76)(58,79)(62,75)(65,74)(68,72)(70,80)(71,77)(73,78);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,14)(15,16)(17,19,23,33)(18,24,26,45)(20,30,34,56)(21,38,41,69)(22,43,39,32)(25,44,49,71)(27,51,50,61)(28,52,42,46)(29,53,55,37)(31,59,60,40)(35,65,62,80)(36,68,63,58)(47,72,73,79)(48,74,67,70)(54,75,76,64)(57,78,77,66);(1,4,5,12,6,13,10,16)(2,7,9,15,3,11,8,14)(17,20,23,34)(18,25,26,49)(19,30,33,56)(21,35,41,62)(22,36,39,63)(24,44,45,71)(27,47,50,73)(28,48,42,67)(29,54,55,76)(31,57,60,77)(32,58,43,68)(37,64,53,75)(38,65,69,80)(40,66,59,78)(46,70,52,74)(51,72,61,79);(2,3)(7,11)(8,9)(14,15)(18,26)(20,34)(24,45)(27,50)(28,42)(29,55)(30,56)(35,62)(36,63)(37,53)(46,52)(51,61)(57,77)(58,68)(65,80)(66,78)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391149);