Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2295 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)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,22,26)(16,20,23,27)(21,25,28,24);(6,9)(10,12)(14,18)(15,19)(16,23)(21,24)(22,26)(25,28);(1,2,3,4)(13,15)(14,19)(16,24)(17,22)(18,26)(20,21)(23,25)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,20,18,27)(15,21,22,28)(19,25,26,24);(13,15)(14,19)(16,25)(17,22)(18,26)(20,28)(21,27)(23,24)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490565);