Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483962 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23 | #442 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18)(19,30,34,60)(20,36)(21,26)(22,43)(23,29)(24,33,49,67)(25,39)(27,41)(28,42)(31,61,64,80)(32,59,35,54)(37,51)(38,55)(40,52)(44,57,68,48)(45,50)(46,53)(47,72,78,76)(56,73)(58,65,77,62)(63,79,66,69)(70,74)(71,75);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(17,19,23,34)(18,24,29,49)(20,31,40,64)(21,32,42,35)(22,44,46,68)(25,47,51,78)(26,48,28,57)(27,54,56,59)(30,43,60,53)(33,41,67,73)(36,69,52,79)(37,62,39,65)(38,72,45,76)(50,61,55,80)(58,75,77,70)(63,74,66,71);(1,4,10,14,6,13,5,12)(2,7,8,15,3,11,9,16)(17,20,21,39,23,40,42,37)(18,25,28,36,29,51,26,52)(19,31,32,65,34,64,35,62)(22,45,41,71,46,38,73,74)(24,47,57,69,49,78,48,79)(27,55,43,75,56,50,53,70)(30,58,54,80,60,77,59,61)(33,66,44,76,67,63,68,72);(2,3)(7,11)(8,9)(15,16)(17,21,23,42)(18,26,29,28)(19,32,34,35)(20,37,40,39)(22,41,46,73)(24,48,49,57)(25,36,51,52)(27,53,56,43)(30,59,60,54)(31,62,64,65)(33,44,67,68)(38,71,45,74)(47,69,78,79)(50,70,55,75)(58,80,77,61)(63,76,66,72)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391160);