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 | C2×#2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,25)(24,28)(26,27);(5,7,9,8)(6,10,12,11)(15,24)(16,18)(19,27)(20,28)(21,23)(25,26);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,15)(14,19)(16,24)(17,25)(18,26)(20,22)(21,27)(23,28);(5,8,9,7)(6,10,12,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,28)(23,27);(13,16,17,18)(14,21,22,23)(15,24,25,26)(19,27,20,28)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=f,e^2=i,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, 6351656);