Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1957 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)(5,6,8,9)(7,10,11,12)(15,19)(18,20)(21,22,27,24)(23,30,28,25)(26,29,35,32)(31,34,36,33);(7,11)(10,12)(14,17)(18,20)(21,23,27,28)(22,25,24,30)(26,31,35,36)(29,33,32,34);(6,9)(10,12)(13,14,16,17)(15,18,19,20)(21,24,27,22)(23,25,28,30)(26,32,35,29)(31,33,36,34);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,25)(22,28)(23,24)(26,34)(27,30)(29,31)(32,36)(33,35);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5517574);