Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2368 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)(5,6,9,11)(7,8,12,10)(13,14,17,19)(15,16,20,18)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,35,36,31);(5,7,9,12)(6,10,11,8)(13,15,17,20)(14,18,19,16)(22,27)(23,28)(25,34)(31,35);(6,11)(7,12)(14,19)(16,18)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,17,18)(14,15,19,20)(21,24)(22,28)(23,27)(25,35)(26,32)(29,30)(31,34)(33,36);(5,8)(6,12)(7,11)(9,10)(13,15)(14,16)(17,20)(18,19)(21,25,26,34)(22,29,27,33)(23,30,28,36)(24,31,32,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2981058);