Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
636 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 36 points:
(5,6,12,17)(7,18)(8,23)(9,27)(10,30)(11,31,29,16)(13,28,33,20)(14,24)(15,19)(21,34,25,32)(22,36)(26,35);(1,2)(3,4)(5,7,10,21)(6,13,15,23)(8,24,20,11)(9,28,29,26)(12,22,19,25)(14,18,16,34)(17,33,30,35)(27,36,31,32);(2,4)(5,8,9,25)(6,13,14,18)(7,19,20,29)(10,28,11,22)(12,26,24,21)(15,23,31,32)(16,34,30,35)(17,33,27,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,f^b=fhi,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56217);