Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8,14,25)(4,7)(5,18,20,27)(6,11)(9,23,26,21)(10,17)(12,22,16,15)(13,28)(19,31)(24,29)(30,32);(1,3)(2,8)(4,16)(5,19)(6,14)(7,22)(9,24)(10,12)(11,25)(13,20)(15,17)(18,31)(21,29)(23,30)(26,32)(27,28);(1,4,17,11,6,10,7,2)(3,12,15,8,14,16,22,25)(5,18,21,26,20,27,23,9)(13,28,30,24,19,31,29,32);(1,5)(2,9)(3,13)(4,18)(6,20)(7,23)(8,24)(10,27)(11,26)(12,28)(14,19)(15,30)(16,31)(17,21)(22,29)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296215);