Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14,17,16)(15,19,20,18)(21,22)(23,25)(24,26)(27,28);(6,10)(7,12)(13,15,17,20)(14,18,16,19)(21,23,26,28)(22,27,24,25);(2,3)(6,10)(7,12)(22,24)(23,28);(1,3,4,2)(5,7,9,12)(6,11,10,8)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,27,28,25);(5,8,9,11)(6,12,10,7)(13,16,17,14)(15,19,20,18)(21,25,26,27)(22,28,24,23)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=g,e^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 | #72316554, #72316557, #72316561, … |
| maximal normal subgroup of | #72316554, #72316557, #72316561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836455);