Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17462 conjugacy classes of subgroups; only the 1006 normal subgroups are shown.
This group has 17462 conjugacy classes of subgroups — too many to draw. Only its 1006 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,25)(18,21)(19,24)(20,26)(22,28)(23,27);(2,3)(10,11)(13,15,18,19)(14,16,21,24)(17,22,26,27)(20,23,25,28);(10,11)(13,16,18,24)(14,19,21,15)(17,23,26,28)(20,27,25,22);(1,3,4,2)(5,6)(7,8)(9,11,12,10)(13,14)(15,16)(17,20)(18,21)(19,24)(22,23)(25,26)(27,28);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,25)(24,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280538 |
| maximal normal subgroup of | #72280538 |
| maximal quotient of | #72306998 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836229);