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 | |
2304 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)(7,11)(10,12)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,28,29,24)(25,30,33,34)(31,35,36,32);(5,6,9,8)(7,10,11,12)(15,20)(17,19)(22,27)(23,29)(25,33)(32,35);(5,7)(6,10)(8,12)(9,11)(13,15,16,20)(14,17,18,19)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,25,26,33)(22,30,27,34)(23,31,29,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fh,d^2=gh,e^2=f,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, 997923);