Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142220 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
705 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 705 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,15)(14,16)(17,18)(19,25,27,22)(20,24)(21,23)(26,30,28,32)(29,31);(1,3,4,8)(2,5,6,7)(19,26,27,28)(22,30,25,32);(10,13)(11,16)(14,15)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)Polycyclic presentation:
2^9|a^2=d,b^2=d,e^2=gi,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,d^c=di,e^c=egi,f^c=fi,h^c=hi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72266618 |
| maximal normal subgroup of | #72266618 |
| maximal quotient of | #72266617, #72267754 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51634);