Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,19,20,17)(21,22,26,25)(23,28,27,24);(6,9)(7,8)(13,14,16,18)(15,17,20,19)(21,23)(22,24)(25,28)(26,27);(5,7,10,8)(6,11,9,12)(13,15)(14,17)(16,20)(18,19)(21,24,26,28)(22,27,25,23);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,24)(25,28)(26,27);(5,9,10,6)(7,11,8,12)(13,15)(14,17)(16,20)(18,19)(21,25,26,22)(23,24,27,28)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3773986);