Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #117624 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,3,4,12,6,11,5,13)(2,7,9,10,8,15,14,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(2,8)(3,11)(4,5)(7,15)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235809 |
| maximal normal subgroup of | #72235809 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 24824);