Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12099 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
841 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 841 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,23,29)(19,28,22,25)(20,26,31,24)(21,32,30,27);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,22)(18,24,29,26)(20,30,31,21)(25,32,28,27);(18,25)(19,22)(21,30)(24,27)(26,32)(28,29);(17,20,23,31)(18,26,29,24)(19,30,22,21)(25,32,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227828, #72227843, #72227847, … |
| maximal normal subgroup of | #72227828, #72227843, #72227847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391533);