Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15763 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2490 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)(6,7)(8,12)(15,19)(18,20)(22,27)(24,31)(29,33)(32,36);(6,7)(10,11)(14,17)(18,20)(21,22,26,27)(23,24,28,31)(25,33,34,29)(30,36,35,32);(5,6,9,7)(8,10,12,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,30)(26,28)(27,31)(29,32)(33,36)(34,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,33)(34,36);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,32,31,36)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374205);