Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153129 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,21)(12,26)(13,17)(14,19)(15,20)(16,23)(18,29)(22,37)(24,38)(25,35)(27,30)(28,34)(31,32)(33,39)(36,40);(1,3,4,8)(2,5,6,7)(9,10)(11,22)(12,27)(13,18)(14,31)(15,20)(16,23)(17,29)(19,32)(21,37)(24,35)(25,38)(26,30)(28,34)(33,40)(36,39);(9,11,15,25)(10,16,20,34)(12,28,13,23)(14,24,32,37)(17,35,26,21)(18,22,27,38)(19,36,31,33)(29,40,30,39)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,e^b=eg,g^b=gi,h^b=hi,e^c=ei,g^c=gh,f^e=fhi,g^e=gi,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60422);