Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1084 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 140 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)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,28)(20,35)(21,37)(23,32)(26,31)(29,36)(30,34)(33,40)(38,39);(2,5)(6,8)(9,11,14,19)(10,15,18,28)(12,20,23,33)(13,21,27,34)(16,29,31,39)(17,30,25,37)(22,35,32,40)(24,36,26,38);(1,3,4,7)(2,6,5,8)(11,19)(15,28)(17,25)(20,33)(21,34)(22,32)(29,39)(36,38);(9,12,14,23)(10,16,18,31)(11,20,19,33)(13,26,27,24)(15,29,28,39)(17,32,25,22)(21,38,34,36)(30,40,37,35)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,c^b=ci,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72303233, #72303234, #72303242, … |
| maximal normal subgroup of | #72303233, #72303234, #72303242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419321);