Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1893 |
| prime radical | #19101 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,3,7,5,8,4,6)(9,10,13,18,15,19,14,29)(11,20,23,36,21,34,24,39)(12,25,26,31,28,40,27,17)(16,30,33,37,32,38,35,22);(2,6)(3,4)(7,8)(9,11,12,22,13,23,26,30,15,21,28,37,14,24,27,38)(10,16,31,34,29,35,25,36,19,32,17,20,18,33,40,39);(2,7)(3,4)(6,8)(11,21)(16,32)(20,34)(22,37)(23,24)(30,38)(33,35)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45843);