Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| 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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(14,18)(16,24)(19,26)(20,27)(21,28)(23,25);(6,9)(7,11)(13,14,16,18,17,19,24,26)(15,21,23,28,22,27,25,20);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(18,26)(23,25);(6,9)(10,12)(13,15,17,22)(14,20,19,28)(16,25,24,23)(18,27,26,21);(5,7,8,11)(6,10,9,12)(14,19)(18,26)(20,28)(21,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056414);