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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2110 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)(7,11)(8,9)(10,12)(13,14,18,20)(15,16,17,19)(22,26)(23,29)(24,32)(34,36);(6,9)(7,12)(13,15)(14,19)(16,20)(17,18)(22,26)(24,32)(27,30)(31,35);(6,9)(10,11)(14,20)(15,17)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,2,3,4)(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,6)(7,10)(8,9)(11,12)(13,17,18,15)(14,16,20,19)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fgh,c^2=fg,d^2=hi,e^2=fgh,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, 4991688);