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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,11)(7,8,12,10)(15,19)(17,20)(21,22,26,30)(23,24,28,27)(25,33,34,29)(31,35,36,32);(2,4)(5,7,9,12)(6,10,11,8)(13,14,16,18)(15,17,19,20)(23,28)(24,27)(31,36)(32,35);(1,2)(3,4)(6,11)(7,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,28)(22,27,30,24)(25,31,34,36)(29,35,33,32);(5,6)(7,8)(9,11)(10,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,36,33,31);(5,8)(6,12)(7,11)(9,10)(21,25,26,34)(22,29,30,33)(23,31,28,36)(24,32,27,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fg,d^2=fg,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, 2993829);