Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #23917 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1098 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(5,6,10,16,11,17,18,35)(7,13,8,21,19,29,23,34)(9,14,26,32,25,15,27,33)(12,28,30,24,31,22,36,20);(1,2)(3,4)(5,7,11,19)(6,12,17,31)(8,10,23,18)(9,24,25,20)(13,15,29,14)(16,30,35,36)(21,33,34,32)(22,26,28,27);(2,4)(5,8,11,23)(6,13,17,29)(7,18,19,10)(9,22,25,28)(12,14,31,15)(16,34,35,21)(20,26,24,27)(30,33,36,32);(6,14)(7,19)(9,25)(10,18)(12,29)(13,31)(15,17)(16,32)(21,36)(22,28)(30,34)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408331);