Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 268435456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2946 conjugacy classes of subgroups; only the 818 normal subgroups are shown.
This group has 2946 conjugacy classes of subgroups — too many to draw. Only its 818 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) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(11,15)(13,16)(17,18,22,25)(19,20,23,28)(21,29,30,24)(26,32,31,27);(5,6,8,7)(9,10,12,14)(11,13,15,16)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,32)(30,31);(1,2,3,4)(5,7,8,6)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,29)(30,32);(9,11,12,15)(10,13,14,16);(17,21,22,30)(18,24,25,29)(19,26,23,31)(20,27,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215806);