Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(5,7,9,8)(6,10,12,11)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,24)(27,28);(1,2,3,4)(5,7,9,8)(6,11,12,10)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(5,8,9,7)(6,10,12,11)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(15,22)(18,26)(21,23)(25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351811);