Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,8)(9,10)(11,12)(14,18)(16,19)(21,22,24,26)(23,27,28,25);(2,4)(5,7,11,10)(6,9,12,8)(14,18)(15,20);(1,2)(3,4)(5,8,11,9)(6,7,12,10)(13,14)(15,19)(16,20)(17,18);(5,9,11,8)(6,7,12,10)(13,15,17,20)(14,19,18,16)(21,23,24,28)(22,25,26,27);(5,10,11,7)(6,9,12,8)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2561098);