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 | |
1918 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)(7,9)(8,10)(11,12)(15,19)(18,20)(21,22,26,30)(23,29,25,27)(24,28,32,35)(31,36,33,34);(5,6,8,10)(7,11,12,9)(14,17)(18,20)(21,23,26,25)(22,27,30,29)(24,31,32,33)(28,34,35,36);(5,6,8,10)(7,9,12,11)(13,14,16,17)(15,18,19,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,36)(30,35);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,23)(22,29,30,27)(24,31,32,33)(28,34,35,36);(5,7,8,12)(6,9,10,11)(21,22)(23,29)(24,28)(25,27)(26,30)(31,36)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=gh,d^2=hi,e^2=g,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, 3045788);