Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2086 conjugacy classes of subgroups; only the 582 normal subgroups are shown.
This group has 2086 conjugacy classes of subgroups — too many to draw. Only its 582 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) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(9,10,13,14)(11,15,16,12)(17,18,20,21)(19,22,24,23)(25,26,28,30)(27,31,32,29);(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,16)(13,14)(18,21)(22,23);(2,6)(4,7)(10,14)(12,15)(17,19,20,24)(18,22,21,23);(1,3,5,8)(2,4,6,7)(9,11,13,16)(10,12,14,15)(17,18)(19,22)(20,21)(23,24)(25,27,28,32)(26,29,30,31);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fh,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227425 |
| maximal subgroup of | #72227425, #72227434 |
| maximal normal subgroup of | #72227425, #72227434 |
| maximal quotient of | #72227423, #72268609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038419);