Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
2267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(13,14,17,19)(15,16,20,18)(22,26)(24,32)(27,29)(30,34);(2,4)(7,11)(10,12)(15,20)(16,18)(21,22,25,26)(23,29,31,27)(24,33,32,28)(30,35,34,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,27)(24,33)(25,26)(28,32)(29,31)(30,36)(34,35);(5,6)(7,10)(8,9)(11,12)(21,23)(22,27)(24,34)(25,31)(26,29)(28,36)(30,32)(33,35);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,15,19,20)(21,24,25,32)(22,28,26,33)(23,30,31,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3992328);