Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1775 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,28,20,31)(21,26,22,32)(24,29,25,30);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,24,27,25)(21,30,22,29)(26,31,32,28);(10,13)(14,16)(17,20,23,19)(18,25,27,24)(21,29,22,30)(26,28,32,31);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,20)(24,25)(28,31)(29,30);(11,15)(14,16)(17,21,23,22)(18,26,27,32)(19,29,20,30)(24,28,25,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6441710);