Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
818 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)(5,6,8,10)(7,9,11,12)(13,14,19,21)(15,16,18,24)(17,20,26,27)(22,23,25,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,19,18)(14,16,21,24)(17,22,26,25)(20,23,27,28);(2,3)(7,11)(9,12)(13,16,18,21,19,24,15,14)(17,23,25,27,26,28,22,20);(1,3,4,2)(13,17,18,25,19,26,15,22)(14,20,16,23,21,27,24,28)Polycyclic presentation:
2^9|a^2=eg,b^2=ei,c^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370515);