Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
2112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,16)(12,14)(13,15)(19,23)(22,24)(25,26,28,29)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,13)(15,16)(26,29)(30,31);(2,5)(6,7)(11,12)(14,16)(17,18,21,20)(19,22,23,24)(25,27)(26,30)(28,32)(29,31);(1,2)(3,7)(4,5)(6,8)(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20,21,18)(19,24,23,22)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=fgh,c^2=fghi,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2667559);