Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508078 |
| 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×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1065 conjugacy classes of subgroups — too many to draw. Only its 124 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)(11,19)(12,24)(13,28)(14,16)(15,21)(17,27)(18,31)(20,37)(22,38)(23,33)(25,32)(26,34)(29,39)(30,36)(35,40);(2,5)(6,8)(9,10)(11,20)(12,17)(13,28)(14,16)(15,22)(18,31)(19,37)(21,38)(23,35)(24,27)(25,39)(26,36)(29,32)(30,34)(33,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,25)(13,29)(14,23)(16,32)(17,33)(18,35)(19,24)(20,28)(21,27)(22,31)(26,37)(30,40)(34,38)(36,39);(3,7)(6,8)(10,16)(15,32)(17,24)(18,28)(19,33)(20,35)(34,36)(38,39)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415278);