Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2828 conjugacy classes of subgroups; only the 554 normal subgroups are shown.
This group has 2828 conjugacy classes of subgroups — too many to draw. Only its 554 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:
(2,3)(4,8)(9,10,14,12)(11,16,15,13)(18,22)(20,23)(27,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,24)(26,29)(30,32);(2,3)(6,7)(10,12)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,16,15,13)(17,19,21,24)(18,23,22,20);(1,4,5,8)(2,6,3,7)(9,13,14,16)(10,15,12,11)(17,20,21,23)(18,24,22,19)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fi,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371485);