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)(19,25,33,50)(20,34,40,67)(21,27,23,45)(22,42,44,53)(26,51,55,38)(28,36,56,43)(30,57,35,64)(31,47,41,49)(32,62,63,73)(37,65,39,72)(46,69,52,60)(48,58,76,61)(54,71,66,70)(59,77,68,78)(74,79,75,80);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,35)(21,41)(22,32)(23,31)(24,33)(26,52)(27,49)(28,48)(29,50)(30,40)(34,64)(36,58)(37,59)(38,69)(39,68)(42,62)(43,61)(44,63)(45,47)(46,55)(51,60)(53,73)(54,74)(56,76)(57,67)(65,77)(66,75)(70,80)(71,79)(72,78);(17,20,36,70,23,39,22,38,24,40,43,71,21,37,44,51)(18,26,53,72,27,54,28,34,29,55,42,65,45,66,56,67)(19,30,58,79,31,59,32,60,33,35,61,80,41,68,63,69)(25,46,73,77,49,75,48,57,50,52,62,78,47,74,76,64);(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)(18,27,29,45)(19,31,33,41)(20,37,40,39)(22,36,44,43)(25,47,50,49)(26,54,55,66)(28,42,56,53)(30,59,35,68)(32,61,63,58)(34,65,67,72)(38,70,51,71)(46,74,52,75)(48,73,76,62)(57,77,64,78)(60,80,69,79)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,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,c^b=ci,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, 7532141);