Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #62437 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,26)(19,29,31,24)(20,34,36,65)(22,41,39,48)(23,43,40,28)(25,47,49,37)(27,35,51,42)(30,55,57,64)(32,62,60,72)(33,46,61,54)(38,71,69,67)(44,58,52,74)(45,63,53,56)(50,70,66,68)(59,80,78,76)(73,77,75,79);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,44)(26,29)(27,45)(28,46)(34,64)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,72)(42,63)(43,54)(47,74)(48,62)(49,52)(50,73)(51,53)(55,65)(66,75)(67,80)(68,77)(69,78)(70,79)(71,76);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,35,68,23,38,22,37,21,36,42,70,40,69,39,47)(18,25,48,71,28,50,27,34,26,49,41,67,43,66,51,65)(19,30,56,77,33,59,32,58,31,57,63,79,61,78,60,74)(24,44,62,76,46,73,45,64,29,52,72,80,54,75,53,55);(3,12)(7,10)(11,13)(15,16)(19,31)(20,36)(24,29)(25,49)(32,60)(33,61)(34,65)(37,47)(38,69)(45,53)(46,54)(50,66)(56,63)(62,72)(67,71)(68,70);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,26)(19,31)(25,49)(27,51)(28,43)(30,57)(32,60)(33,61)(34,65)(41,48)(50,66)(56,63)(58,74)(59,78)(67,71)(77,79)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532324);