Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,11)(8,9)(10,12)(14,18)(15,16)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,22,25,28)(23,29,31,26)(24,27,33,36)(30,35,32,34);(1,2,3,4)(5,7,9,12)(6,10,8,11)(13,15,17,16)(14,19,18,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7)(6,11)(8,10)(9,12)(13,14)(15,20)(16,19)(17,18)(21,23,25,31)(22,26,28,29)(24,32,33,30)(27,35,36,34);(5,8,9,6)(7,11,12,10)(13,16,17,15)(14,20,18,19)(21,24,25,33)(22,27,28,36)(23,30,31,32)(26,34,29,35)Polycyclic presentation:
2^9|b^2=fg,c^2=hi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2649399);