Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144135 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 96 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)(11,21)(12,18)(13,31)(14,32)(15,20)(16,23)(17,28)(19,29)(22,34)(24,39)(25,36)(26,40)(27,38)(30,37)(33,35);(2,6)(3,4)(7,8)(9,11)(10,16)(12,27)(13,22)(14,33)(15,24)(17,35)(18,21)(19,38)(20,34)(23,28)(25,30)(26,29)(31,40)(32,36)(37,39);(2,7)(3,4)(6,8)(11,22)(12,28)(13,30)(14,29)(16,21)(17,18)(19,32)(23,24)(25,33)(26,27)(31,37)(34,39)(35,38)(36,40)Polycyclic presentation:
2^9|d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fghi,g^b=gi,d^c=dg,e^c=eh,f^c=fghi,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224858 |
| maximal normal subgroup of | #72224858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51331);