Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489719 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 687 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,24,30)(19,31,35,59)(20,26,38,52)(21,29,23,27)(22,41,44,54)(25,46,49,34)(28,40,56,43)(32,58,63,79)(33,64,45,55)(36,53,39,50)(37,66,68,77)(42,57,70,48)(47,71,74,61)(51,69,76,65)(60,75,62,78)(67,72,80,73);(4,13)(8,12)(9,16)(11,14)(17,19,24,35)(18,25,30,49)(20,32,38,63)(21,33,23,45)(22,42,44,70)(26,47,52,74)(27,48,29,57)(28,55,56,64)(31,54,59,41)(34,40,46,43)(36,60,39,62)(37,67,68,80)(50,72,53,73)(51,75,76,78)(58,77,79,66)(61,69,71,65);(1,3,5,10)(2,6,7,15)(4,14,13,11)(8,16,12,9)(17,20)(18,26)(19,32)(21,39)(22,37)(23,36)(24,38)(25,47)(27,53)(28,51)(29,50)(30,52)(31,58)(33,62)(34,61)(35,63)(40,69)(41,66)(42,67)(43,65)(44,68)(45,60)(46,71)(48,73)(49,74)(54,77)(55,75)(56,76)(57,72)(59,79)(64,78)(70,80);(2,7)(6,15)(8,12)(9,16)(17,21,24,23)(18,27,30,29)(19,33,35,45)(20,36,38,39)(22,40,44,43)(25,48,49,57)(26,50,52,53)(28,54,56,41)(31,55,59,64)(32,60,63,62)(34,42,46,70)(37,65,68,69)(47,72,74,73)(51,66,76,77)(58,78,79,75)(61,80,71,67)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397042);