Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
2300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,8,12,10)(13,14)(15,16)(17,19)(18,20)(21,22,26,28)(23,25,27,24);(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,18,19,16)(21,23,26,27)(22,24,28,25);(6,11)(8,10)(15,20)(16,18)(21,24,26,25)(22,27,28,23);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,19)(18,20)(21,25,26,24)(22,27,28,23);(5,8,9,10)(6,7,11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgh,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2227469);