Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3319 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2120 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)(5,6,8,10)(7,9,11,12)(13,14)(15,22)(16,24)(17,21)(18,19)(20,26)(23,27)(25,28);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,18)(14,19,21,22)(16,25,26,23)(20,28,24,27);(7,11)(9,12)(14,19)(15,18)(16,26)(20,28)(21,22)(24,27);(5,7)(6,9)(8,11)(10,12)(13,16,17,26)(14,20,21,24)(15,23,18,25)(19,27,22,28);(7,11)(9,12)(16,26)(20,24)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498823);