Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 | #2215 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,24,29)(19,25,33,49)(20,34,39,63)(21,27,23,44)(22,42,43,69)(26,50,52,37)(28,41,53,35)(30,54,58,78)(31,46,40,48)(32,59,60,74)(36,61,38,67)(45,70,73,57)(47,68,75,55)(51,64,62,66)(56,76,65,77)(71,80,72,79);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,30)(21,40)(22,32)(23,31)(24,33)(26,45)(27,48)(28,47)(29,49)(34,54)(35,55)(36,65)(37,57)(38,56)(39,58)(41,68)(42,59)(43,60)(44,46)(50,70)(51,72)(52,73)(53,75)(61,77)(62,71)(63,78)(64,79)(66,80)(67,76)(69,74);(17,20,35,64,23,38,43,50,24,39,41,66,21,36,22,37)(18,26,42,61,27,51,53,63,29,52,69,67,44,62,28,34)(19,30,55,79,31,56,60,70,33,58,68,80,40,65,32,57)(25,45,59,77,48,72,75,78,49,73,74,76,46,71,47,54);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,21,24,23)(18,27,29,44)(19,31,33,40)(20,36,39,38)(22,41,43,35)(25,46,49,48)(26,51,52,62)(28,42,53,69)(30,56,58,65)(32,55,60,68)(34,61,63,67)(37,66,50,64)(45,71,73,72)(47,74,75,59)(54,76,78,77)(57,79,70,80)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,d^2=f,e^2=i,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=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532134);