Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
2345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,34)(35,36);(5,7)(6,11)(8,9)(10,12)(13,14,16,18)(15,17,20,19)(22,27)(23,28)(29,36)(30,33);(6,8)(7,12)(13,15,16,20)(14,17,18,19)(21,23,26,28)(22,24,27,32)(25,33,35,30)(29,34,36,31);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,15)(14,17)(16,20)(18,19)(21,24)(22,28)(23,27)(25,34)(26,32)(29,30)(31,35)(33,36);(5,9)(6,7)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,25,26,35)(22,29,27,36)(23,30,28,33)(24,31,32,34)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=gh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4052532);