Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1770 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)(6,9)(10,12)(13,14)(15,20)(16,18)(17,19)(21,22,26,30)(23,28,33,27)(24,25,35,29)(31,32,34,36);(7,11)(10,12)(13,15)(14,19)(16,20)(17,18)(21,23,26,33)(22,27,30,28)(24,34,35,31)(25,32,29,36);(5,6,8,9)(7,10,11,12)(13,14,18,16)(15,19,17,20)(21,23)(22,28)(24,34)(25,36)(26,33)(27,30)(29,32)(31,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,17,20)(21,24)(22,29)(23,31)(25,30)(26,35)(27,32)(28,36)(33,34);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,19,16,20)(21,25,26,29)(22,24,30,35)(23,32,33,36)(27,34,28,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4991259);