Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | |
2318 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)(5,6,10,12)(7,8,11,9)(13,14,16,18)(15,19,20,17)(21,22)(23,30)(24,28)(25,33)(26,27)(29,34)(31,32)(35,36);(5,7,10,11)(6,8,12,9)(15,20)(17,19)(22,27)(23,28)(25,34)(32,36);(5,8,10,9)(6,7,12,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,36)(34,35);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,24)(22,28)(23,27)(25,32)(26,30)(29,35)(31,33)(34,36);(5,9,10,8)(6,11,12,7)(13,14)(15,17)(16,18)(19,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,33)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1004922);