Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114428 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,15,18,17)(16,19,23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,19)(12,17)(13,15)(14,18)(16,20)(21,24)(22,23);(5,7)(9,11)(10,16)(12,20)(13,22)(14,21)(15,19)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210545, #72213643, #72213681, … |
| maximal normal subgroup of | #72210545, #72213643, #72213681, … |
| maximal quotient of | C2×#114428, #72260959, #72261280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21624);