Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 20971520 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3051 conjugacy classes of subgroups; only the 510 normal subgroups are shown.
This group has 3051 conjugacy classes of subgroups — too many to draw. Only its 510 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:
(2,5)(6,8)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24)(26,28)(27,31);(9,10,12,13)(11,14,15,16)(17,19,20,24)(18,21,22,23)(25,26,29,28)(27,32,31,30);(11,15)(14,16)(18,22)(21,23)(26,28)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,21)(20,24)(22,23)(25,27,29,31)(26,30,28,32);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24)(25,28,29,26)(27,30,31,32)Polycyclic presentation:
2^9|b^2=f,d^2=h,e^2=fhi,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 | #72235191, #72235193, #72237378 |
| maximal subgroup of | #72234508, #72234510, #72235148, … |
| maximal normal subgroup of | #72234508, #72234510, #72235148, … |
| maximal quotient of | #72334849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038421);