Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,19)(8,11,9,13)(12,17,20,16)(21,22,26,23,25,28,33,30)(24,27,32,29,31,34,36,35);(2,4)(6,11)(8,9)(12,19)(13,14)(15,20)(16,17)(21,23,26,28,25,30,33,22)(24,29,32,34,31,35,36,27);(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,15)(14,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(5,8,10,9)(6,13,14,11)(7,16,18,17)(12,15,20,19)(22,28)(23,30)(24,31)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dhi,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99221);