Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508066 |
| 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 | #23903 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1189 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)(11,15)(12,22)(13,18)(14,17)(16,23)(19,33)(20,29)(21,28)(24,36)(25,32)(26,31)(27,34)(30,37)(35,40)(38,39);(2,5)(6,8)(9,11,14,21)(10,15,17,28)(12,19,24,35)(13,25,26,20)(16,27,30,39)(18,32,31,29)(22,33,36,40)(23,34,37,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,34)(21,35)(25,38)(26,37)(28,39)(29,33)(31,36)(32,40);(3,7)(6,8)(10,17)(15,28)(16,30)(18,31)(22,36)(27,39)(29,32)(33,40)Polycyclic presentation:
2^9|b^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72341134 |
| maximal normal subgroup of | #72341134 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415263);