Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,8)(6,7)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,2,4,6)(3,7,8,5);(2,6)(5,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(2,7)(3,8)(5,6)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,g^c=gi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127605);