Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,12,16,13,15,21,23)(11,17,19,24,18,14,20,22);(9,11,13,18)(10,14,15,17)(12,20,21,19)(16,24,23,22);(1,2,4,6)(3,8,7,5)(14,17)(22,24);(10,15)(12,21)(14,17)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fi,g^d=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122004);