Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(10,11)(13,14,18,22,17,23,19,24)(15,25,26,21,16,20,27,28);(5,6)(7,8)(9,10,12,11)(13,15,17,16)(14,20,23,25)(18,26,19,27)(21,22,28,24);(2,4)(6,8)(13,16,17,15)(14,21,23,28)(18,26,19,27)(20,22,25,24);(9,11,12,10)(14,22)(18,19)(20,28)(21,25)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303498);