Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,16)(13,28)(14,17)(15,21)(18,25)(19,26)(22,39)(23,36)(24,35)(27,32)(29,34)(30,38)(31,40)(33,37);(1,2,4,6)(3,8,7,5)(9,11,14,23)(10,15,17,30)(12,24,27,40)(13,29,26,22)(16,20,32,36)(18,34,33,39)(19,35,28,31)(21,37,38,25);(10,16)(12,25)(13,26)(15,20)(17,32)(18,19)(21,24)(22,29)(27,37)(28,33)(30,36)(31,39)(34,35)(38,40);(10,17)(15,30)(16,32)(18,33)(19,28)(20,36)(31,35)(34,39)Polycyclic presentation:
2^9|b^2=gh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309481, #72309493, #72309504, … |
| maximal normal subgroup of | #72309481, #72309493, #72309504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396231);