Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501144 |
| 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 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1081 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1081 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:
(1,2,5,12,7,13,6,9)(3,14,17,30,18,29,15,31)(4,11,21,28,20,10,22,27)(8,23,25,16,24,32,26,19);(1,3)(2,8)(4,19)(5,17)(6,15)(7,18)(9,25)(10,14)(11,29)(12,26)(13,24)(16,20)(21,23)(22,32)(27,31)(28,30);(2,9)(3,15)(4,20)(5,6)(8,24)(10,27)(11,28)(12,13)(14,29)(16,23)(17,18)(19,32);(2,10)(4,20)(8,14)(9,27)(11,13)(12,28)(16,19)(21,22)(23,32)(24,29)(25,31)(26,30)Polycyclic presentation:
2^9|a^2=f,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,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408340);