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 | #537 | C23 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
770 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,30)(21,25)(22,32)(23,28)(24,29)(26,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,22)(18,24,28,30)(20,31,29,21)(25,27,26,32);(17,20,23,29)(18,25,28,26)(19,21,22,31)(24,32,30,27);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gi,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 | #72266670 |
| maximal quotient of | #72266618 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391690);