Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | |
2169 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)(8,10)(9,12)(14,17)(15,19)(21,22)(23,26)(24,29)(25,30)(27,31)(28,33)(32,34)(35,36);(5,7,10,9)(6,11,8,12)(15,19)(18,20)(22,23)(24,31)(25,33)(34,35);(5,6,10,8)(7,12,9,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,34,36,35);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,15,16,19)(14,18,17,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9,10,7)(6,11,8,12)(13,14)(15,18)(16,17)(19,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|b^2=fh,c^2=h,d^2=fi,e^2=fh,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, 1423974);