Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501150 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #23872 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1077 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1077 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,14,7,13)(3,15,19,28,20,27,16,10)(4,21,23,12,24,30,5,9)(17,29,32,31,22,25,18,26);(2,9)(3,16)(5,23)(6,7)(10,26)(11,21)(12,14)(13,30)(15,29)(17,32)(18,22)(19,20)(25,27)(28,31);(1,3,8,20)(2,10,14,28)(4,22,24,17)(5,18,23,32)(6,16,7,19)(9,25,12,29)(11,27,13,15)(21,31,30,26);(1,4,8,24)(2,11,14,13)(3,17,20,22)(5,7,23,6)(9,21,12,30)(10,15,28,27)(16,32,19,18)(25,26,29,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=gi,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, 408350);