Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2001 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)(13,14)(15,18)(16,17)(19,20)(21,22,27,31)(23,32,24,28)(25,34,35,29)(26,33,36,30);(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,23,27,24)(22,28,31,32)(25,30,35,33)(26,34,36,29);(5,6,8,10)(7,9,11,12)(13,15,17,20)(14,19,16,18)(21,24,27,23)(22,28,31,32)(25,33,35,30)(26,34,36,29);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,18)(17,20)(21,25,27,35)(22,29,31,34)(23,33,24,30)(26,28,36,32);(5,7)(6,9)(8,11)(10,12)(13,16,17,14)(15,18,20,19)(21,26,27,36)(22,30,31,33)(23,34,24,29)(25,28,35,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5190642);