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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(7,11)(13,14,17,16)(15,19,20,18)(22,27)(25,28);(2,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22,26,27)(23,28,24,25);(1,2,3,4)(6,8)(10,12)(14,16)(18,19)(21,23,26,24)(22,25,27,28);(5,7,9,11)(6,10,8,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,28,27,25);(5,8,9,6)(7,10,11,12)(13,16,17,14)(15,18,20,19)(21,25,26,28)(22,24,27,23)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=hi,d^2=fg,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, 3851403);