Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| 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 | #3485 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
947 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 947 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,21,29)(12,22,24,16)(13,25,26,17)(19,27,32,37)(20,33,34,28)(23,30,36,35)(31,39,40,38);(1,3,4,8)(2,5,6,7)(11,19)(13,26)(15,27)(17,25)(20,31)(21,32)(23,36)(28,38)(29,37)(30,35)(33,39)(34,40);(9,11)(10,15)(12,20)(13,19,26,32)(14,21)(16,28)(17,27,25,37)(18,29)(22,33)(23,31,36,40)(24,34)(30,38,35,39);(2,6)(5,7)(9,12,14,24)(10,16,18,22)(11,20,21,34)(13,23,26,36)(15,28,29,33)(17,30,25,35)(19,31,32,40)(27,38,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409866);