Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
2260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,10)(7,12)(13,14,17,20)(15,18,16,19)(22,25)(26,27);(1,2,3,4)(5,6,9,10)(7,11,12,8)(21,22,24,25)(23,27,28,26);(1,2)(3,4)(13,15,17,16)(14,18,20,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,19)(17,20)(21,23,24,28)(22,26,25,27);(5,8,9,11)(6,12,10,7)(13,16,17,15)(14,19,20,18)(21,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=fgh,c^2=fgi,d^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2929272);