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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1896 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)(7,11,12,10)(13,14)(15,16)(17,18)(19,20)(21,22)(23,31)(24,34)(25,30)(26,27)(28,32)(29,35)(33,36);(6,9)(7,12)(13,14,17,18)(15,19,20,16)(22,27)(23,32)(24,35)(25,36);(6,9)(10,11)(13,15,17,20)(14,16,18,19)(21,23)(22,28)(24,33)(25,34)(26,32)(27,31)(29,36)(30,35);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,29)(23,33)(25,31)(26,35)(27,34)(28,36)(30,32);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)(21,25,26,36)(22,30,27,33)(23,29,32,34)(24,31,35,28)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fg,d^2=gi,e^2=fgh,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, 5190450);