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 | |
1930 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)(8,12)(9,10)(13,14,17,18)(15,19,20,16)(21,22)(23,28)(24,33)(25,31)(26,27)(29,34)(30,36)(32,35);(5,7,9,12)(6,8,10,11)(13,14)(15,19)(16,20)(17,18)(22,27)(24,34)(28,35)(30,31);(6,10)(8,11)(14,18)(16,19)(21,23,26,32)(22,28,27,35)(24,30,34,31)(25,33,36,29);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,29)(23,31)(25,35)(26,34)(27,33)(28,36)(30,32);(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,25,26,36)(22,30,27,31)(23,29,32,33)(24,35,34,28)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fg,d^2=fi,e^2=fg,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, 5186359);