Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384259 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2333 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14452 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
802 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)(3,11,15,7)(4,8,17,24)(5,9)(6,19)(10,12)(13,21)(14,22,20,26)(16,30)(18,32,28,25)(23,31)(27,29);(1,3,12,14,5,15,6,20)(2,7,10,26,9,11,19,22)(4,16,28,29,17,31,18,13)(8,23,25,21,24,30,32,27);(1,4,6,18,5,17,12,28)(2,8,19,32,9,24,10,25)(3,13,14,16,15,29,20,31)(7,21,26,30,11,27,22,23);(2,9)(8,24)(10,19)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=eg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294377);