Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143219 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
553 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 553 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,20)(14,19)(15,21)(18,25)(22,37)(23,39)(24,35)(26,30)(27,31)(28,32)(29,34)(33,38)(36,40);(2,6)(3,4)(7,8)(10,16)(11,22)(12,25)(13,14)(17,33)(18,20)(19,31)(21,32)(23,26)(24,29)(27,28)(30,34)(35,36)(37,39);(2,7)(3,4)(6,8)(9,11)(10,17)(12,26)(13,29)(14,22)(15,24)(16,30)(18,36)(19,37)(20,34)(21,35)(23,28)(25,40)(27,38)(31,33)(32,39)Polycyclic presentation:
2^9|d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50415);