Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220244 |
| 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 | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(2,6)(3,8)(4,7)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24);(10,15)(12,19)(14,20)(17,23);(1,3,5,6)(2,7,8,4)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 127562);