Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,7)(9,10)(11,12)(13,14)(15,23)(16,21)(17,19)(18,22)(20,25)(24,27)(26,28);(1,2,4,3)(5,7,8,6)(14,19)(15,24)(16,18)(20,27)(23,28)(25,26);(1,3,4,2)(6,7)(13,15,17,25)(14,20,19,23)(16,24,18,26)(21,28,22,27);(10,12)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,28,23,27)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304369);