Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103161 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
1006 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(2,6)(3,4)(7,8)(13,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72257807, #72264154, #72264162 |
| maximal normal subgroup of | #72257807, #72264154, #72264162 |
| maximal quotient of | #72264968 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10370);