Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511775 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2416 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 2416 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,30)(18,31)(19,35)(20,27)(21,28)(22,36)(23,29)(24,32)(25,34)(26,33);(5,7,10,18)(6,11,14,25)(8,19,21,16)(9,17,23,34)(12,26,28,24)(13,15,29,31)(20,33,22,32)(27,35,36,30);(5,8,10,21)(6,12,14,28)(7,16,18,19)(9,22,23,20)(11,24,25,26)(13,27,29,36)(15,30,31,35)(17,33,34,32);(2,4)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72261344 |
| maximal normal subgroup of | #72261344 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418973);