Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1523 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)(3,7)(4,8)(5,6)(17,18,23,24)(19,27,28,31)(20,30,32,29)(21,25,22,26);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,30)(22,29)(23,28)(24,32)(25,27)(26,31);(10,13)(14,16)(17,20)(18,19)(21,31)(22,27)(23,32)(24,28)(25,29)(26,30);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,24)(19,28)(25,26)(29,30);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,25,24,26)(19,29,28,30)(20,31,32,27)Polycyclic presentation:
2^9|a^2=i,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6441947);