Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,21,29)(12,16,22,30)(13,24,25,17)(19,27,26,32)(20,28,35,40)(23,37,36,31)(33,38,34,39);(2,5)(6,8)(11,19)(12,22)(13,25)(15,27)(16,30)(17,24)(20,34)(21,26)(28,39)(29,32)(33,35)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,28)(17,32)(18,29)(19,25)(22,35)(23,34)(24,27)(30,40)(31,39)(33,36)(37,38);(2,5)(6,8)(9,12,14,22)(10,16,18,30)(11,20,21,35)(13,23,25,36)(15,28,29,40)(17,31,24,37)(19,33,26,34)(27,38,32,39)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72303059, #72303227, #72303234, … |
| maximal normal subgroup of | #72303059, #72303227, #72303234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407376);