Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2129 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)(7,8)(10,11)(13,14,17,18)(15,16,19,20)(21,22,26,27)(23,28,29,24)(25,30,33,34)(31,35,36,32);(5,6,9,12)(7,10,8,11)(13,15,17,19)(14,16,18,20)(22,27)(23,29)(25,33)(32,35);(5,7,9,8)(6,10,12,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,8,9,7)(6,11,12,10)(13,14)(15,16)(17,18)(19,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fgh,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1555475);