Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 67108864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×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 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14)(15,20)(16,19)(17,18)(21,22,26,23)(24,28,25,27);(1,2,3,4)(6,9)(10,12)(14,18)(19,20)(21,23,26,22)(24,28,25,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24,26,25)(22,27,23,28);(13,15,17,16)(14,19,18,20)(21,25,26,24)(22,28,23,27);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^9|a^2=g,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, 4215803);