Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| 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 | #3496 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 132 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:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,37)(20,29)(21,30)(22,31)(23,36)(24,32)(25,35)(26,33)(27,34)(28,38)(39,40);(1,3,5,4)(2,6,8,7)(9,11,14,20)(10,15,17,29)(12,22,24,39)(13,25,27,21)(16,31,32,40)(18,35,34,30)(19,26,38,23)(28,36,37,33);(1,3,5,4)(2,7,8,6)(9,12)(10,16)(11,19,20,38)(13,26,27,23)(14,24)(15,28,29,37)(17,32)(18,36,34,33)(21,39)(22,25)(30,40)(31,35);(1,4,5,3)(2,7,8,6)(10,17)(15,29)(16,32)(18,34)(28,37)(30,35)(31,40)(33,36)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=ef,d^2=e,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, 410780);