Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,22)(19,25)(24,29)(27,30);(1,2)(3,4)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,22)(21,29)(23,27)(24,28)(25,32)(30,31);(10,14)(13,15)(17,19,20,25)(18,23,22,31)(21,30,28,27)(24,32,29,26);(1,3,4,2)(11,16)(13,15)(18,22)(23,31)(24,29)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292219);